Computer Graphics: the mathematics behind the scenes1024437/FULLTEXT01.pdf · Preface This work is...

2004:15

C/D-UPPSATS

COMPUTER GRAPHICSThe mathematics behind the scenes

Department of MathematicsSupervisor: Thomas Gunnarsson

2004:15 • ISSN: 1402–1781 • ISRN: LTU-C/DUPP--04/15--SE

LENNART JORDANSSON

Abstract

This work describes computer graphics with focus on the mathematics behindthe scenes. Representations of three-dimensional objects, such as curves, sur-faces, and solids are described. Integral and rational Bezier curves, B-splinesand NURBS are used for the modeling of curves and surfaces. The objectscan be visualized as projections, and a useful model for defining the view of ascene, the synthetic-camera model, is described. Basic algorithms for renderingof lines and circles are discussed. Examples of interactive programs in Javawhich demonstrate rendering of lines, circles, ellipses, and Bezier curves, aregiven.

Sammanfattning

Denna C/D-uppsats handlar om matematiken bakom datorgrafik. Matematiskbeskrivning av tredimensionella objekt, sasom kurvor, ytor och kroppar, be-handlas. For modellering av kurvor och ytor anvands Bezierkurvor, B-splinesoch NURBS. Objekten kan visualiseras med hjalp av projektioner. En anvand-bar modell for att definiera vyer forklaras. Grundlaggande algoritmer for rast-rering av linjer och cirklar beskrivs. Interaktiva program skrivna i Java demon-strerar rastrering av linjer, cirklar, ellipser och Bezierkurvor.

Preface

This work is an extended essay in mathematics and concludes the coursesMAM603/604 at Lulea University of Technology. The reader is assumed tohave a basic knowledge of mathematics at a university level, like calculus, linearalgebra and geometry. The paper was written using LATEX and most of thefigures were produced by the mathematics program Maple.

I would like to thank my supervisor Thomas Gunnarsson at the Department

of Mathematics for help and valuable opinions during the work, both regardingthe mathematics, and report writing.

Lulea, May 27, 2004Lennart Jordansson

i

Contents

Introduction 1

1 Mathematical Representation of 3D Objects 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Representation of 3D Curves . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Explicit and Implicit Equations . . . . . . . . . . . . . . . 31.2.2 Parametric Cubic Curves . . . . . . . . . . . . . . . . . . 51.2.3 Bezier Curves . . . . . . . . . . . . . . . . . . . . . . . . 141.2.4 Rational Bezier Curves . . . . . . . . . . . . . . . . . . . 351.2.5 B-Spline Curves . . . . . . . . . . . . . . . . . . . . . . . 381.2.6 B-Spline Curve Types . . . . . . . . . . . . . . . . . . . . 431.2.7 NURBS Curves . . . . . . . . . . . . . . . . . . . . . . . . 441.2.8 Linear Approximations of Curves . . . . . . . . . . . . . . 47

1.3 Representation of 3D Surfaces . . . . . . . . . . . . . . . . . . . . 481.3.1 Explicit and Implicit Equations . . . . . . . . . . . . . . . 481.3.2 Parametric Bicubic Surfaces . . . . . . . . . . . . . . . . . 491.3.3 Rational Bezier Surfaces . . . . . . . . . . . . . . . . . . . 511.3.4 NURBS Surfaces . . . . . . . . . . . . . . . . . . . . . . . 521.3.5 Approximating Surfaces by Polygon Meshes . . . . . . . . 55

1.4 Representation of 3D Solids . . . . . . . . . . . . . . . . . . . . . 571.4.1 Parametric Tricubic Solids . . . . . . . . . . . . . . . . . . 571.4.2 Solids Bounded by Surfaces . . . . . . . . . . . . . . . . . 58

2 Visualizing 3D Objects as 2D Projections 59

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.2 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . 592.2.2 Rotation about a Coordinate Axis . . . . . . . . . . . . . 592.2.3 Scaling about the Origin . . . . . . . . . . . . . . . . . . . 612.2.4 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . 612.2.5 Homogeneous Transformations . . . . . . . . . . . . . . . 622.2.6 Rotation about an Arbitrary Axis . . . . . . . . . . . . . 64

2.3 Planar Geometric Projections . . . . . . . . . . . . . . . . . . . . 672.3.1 Parallel Projections . . . . . . . . . . . . . . . . . . . . . 682.3.2 Perspective Projections . . . . . . . . . . . . . . . . . . . 69

iii

2.4 Viewing in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.4.1 The Synthetic-Camera Model . . . . . . . . . . . . . . . . 732.4.2 View Volume . . . . . . . . . . . . . . . . . . . . . . . . . 752.4.3 Clipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.5 Hidden Surface Removal . . . . . . . . . . . . . . . . . . . . . . . 782.5.1 Back Face Culling . . . . . . . . . . . . . . . . . . . . . . 782.5.2 Depth Buffer Algorithm . . . . . . . . . . . . . . . . . . . 78

3 Rendering the Projected Objects 79

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.2 Rendering Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.2.1 Rendering Lines . . . . . . . . . . . . . . . . . . . . . . . 793.2.2 Rendering Circles . . . . . . . . . . . . . . . . . . . . . . . 833.2.3 Rendering Bezier Curves . . . . . . . . . . . . . . . . . . . 86

3.3 Rendering Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4 Programming Examples 88

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.2 Midpoint Algorithm for Lines, Circles and Ellipses . . . . . . . . 88

4.2.1 Midpoint Line Algorithm . . . . . . . . . . . . . . . . . . 884.2.2 Midpoint Circle Algorithm . . . . . . . . . . . . . . . . . 924.2.3 Midpoint Ellipse Algorithm . . . . . . . . . . . . . . . . . 95

4.3 Algorithm for Bezier Curve . . . . . . . . . . . . . . . . . . . . . 98

Conclusion 104

Bibliography 105

Biography 106

iv

List of Figures

1.1 Rotated and translated ellipse . . . . . . . . . . . . . . . . . . . . 51.2 Parametric cubic curve . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Blending functions F1 and F2 . . . . . . . . . . . . . . . . . . . . 91.4 Blending functions F3 and F4 . . . . . . . . . . . . . . . . . . . . 91.5 Parametric cubic curve . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Reparametrization . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 Composite curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.8 Planar cubic Bezier curve . . . . . . . . . . . . . . . . . . . . . . 151.9 Bernstein polynomials of degree 3 . . . . . . . . . . . . . . . . . . 161.10 Convex hull of points in a plane and in space . . . . . . . . . . . 201.11 Closed planar Bezier curve of degree 5 . . . . . . . . . . . . . . . 211.12 Planar cubic Bezier curves: p1 is changed, direction at u = 0 is

not changed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.13 Planar cubic Bezier curves: p1 is changed, direction at u = 0 is

changed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.14 Closed planar Bezier curve of degree 6 with a multiple point at

(6, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.15 Subdivided cubic Bezier curve . . . . . . . . . . . . . . . . . . . . 361.16 Planar quadratic rational Bezier curve . . . . . . . . . . . . . . . 371.17 Quadratic planar B-spline curve . . . . . . . . . . . . . . . . . . . 401.18 Cubic planar B-Spline curves, p2 is changed . . . . . . . . . . . . 421.19 Closed periodic cubic planar B-spline curve . . . . . . . . . . . . 451.20 Unit circle as a quadratic planar NURBS curve . . . . . . . . . . 461.21 Linear approximation of a cubic Bezier curve . . . . . . . . . . . 481.22 Rotated and translated ellipsoid . . . . . . . . . . . . . . . . . . . 501.23 Rational bicubic Bezier surface . . . . . . . . . . . . . . . . . . . 521.24 Unit sphere created as a NURBS surface . . . . . . . . . . . . . . 531.25 Torus created as a NURBS surface . . . . . . . . . . . . . . . . . 541.26 Surface to be approximated . . . . . . . . . . . . . . . . . . . . . 561.27 Surface approximated by a polygon mesh . . . . . . . . . . . . . 561.28 Polygon mesh with polygon P1,1 triangulated . . . . . . . . . . . 57

2.1 Positive rotation about the z-axis . . . . . . . . . . . . . . . . . . 602.2 Reflections in the xy-plane, in the z-axis, and in the origin . . . . 622.3 Rotation about an arbitrary axis . . . . . . . . . . . . . . . . . . 652.4 Axis of rotation translated . . . . . . . . . . . . . . . . . . . . . . 66

v

2.5 Planar geometric projection . . . . . . . . . . . . . . . . . . . . . 672.6 Parallel projection . . . . . . . . . . . . . . . . . . . . . . . . . . 682.7 Perspective projection . . . . . . . . . . . . . . . . . . . . . . . . 702.8 Synthetic-camera . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.9 View volume for orthogonal projection . . . . . . . . . . . . . . . 762.10 View volume for perspective projection . . . . . . . . . . . . . . . 762.11 Prewarping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.1 Midpoint line algorithm . . . . . . . . . . . . . . . . . . . . . . . 803.2 Choices for current and next pixels . . . . . . . . . . . . . . . . . 813.3 Midpoint circle algorithm . . . . . . . . . . . . . . . . . . . . . . 843.4 Choices for current and next pixels . . . . . . . . . . . . . . . . . 843.5 Approximation of a sphere by polygons . . . . . . . . . . . . . . 87

vi

Introduction

The paper starts by describing how graphical three-dimensional objects can berepresented in a computer. Chapter 1 describes curves, surfaces and solids. Thecurves and surfaces are modeled as parametric functions. Bezier curves and sur-faces are described, as well as B-spline/NURBS curves and surfaces. Chapter 2discusses how the objects can be visualized as projections, and how to determinewhich objects in a scene should be seen. A useful model for determining a view,the synthetic-camera model is described. In Chapter 3, we show some basicalgorithms for rendering lines and circles. Finally, in Chapter 4, some programexamples written in Java are shown. The programs are interactive, and let theuser explore how the algorithms for rendering work.

1

Chapter 1

Mathematical

Representation of 3D

Objects

1.1 Introduction

This chapter describes how graphical 3-dimensional objects can be representedin a computer. The representations of curves are described in Section 1.2,and the representations of surfaces and solids in Section 1.3 and Section 1.4respectively.

1.2 Representation of 3D Curves

This section shows how curves in 3 dimensions can be expressed by paramet-ric equations. Common special cases, as parametric cubic (pc) curves, Beziercurves, and B-spline curves are described.

1.2.1 Explicit and Implicit Equations

For a point (x, y, z) on a curve, the coordinates y and z can be expressed withtwo real-valued functions of x as

y = f(x)

z = g(x) (1.1)

where a ≤ x ≤ b. This formulation with explicit equations has the drawbackthat a curve with, for example, multiple values of y for the same value of x hasto be broken into multiple curve segments, such that for each segment, we geta single value of y and z respectively, for each value of x. For example, a circle

3

(in the z = 0 plane) with radius r, centered at the origin, can be divided intotwo half circles C1 and C2 as:

C1 : y =√

r2 − x2

z = 0 , −r ≤ x ≤ r

C2 : y = −√

r2 − x2

z = 0

(1.2)

A planar curve can be represented by an implicit equation of the form

f(x, y) = 0 (1.3)

In this way, the circle from the example above can be expressed as the solutionsto the equation

x2 + y2 − r2 = 0 (1.4)

To express only the half circle where y ≥ 0, this constraint must be added tothe equation above.

As another example, the general second-degree implicit equation

Ax2 + 2Bxy + Cy2 + Dx + Ey + F = 0 (1.5)

can represent lines and conics (ellipses, parabolas, and hyperbolas), by choosingthe coefficients A, B, C, D, E, and F appropriately. If A, B, and C are all 0,and at least one of D and E is non-zero, the equation represents a line. If atleast one of A, B, and C is non-zero, we have the following cases:

AC − B2 > 0 An ellipse, a point, or the empty set.

AC − B2 < 0 A hyperbola, or two crossing lines.

AC − B2 = 0 A parabola, two parallel lines, a single line, or the empty set.

As an example of the first case, consider the equation

7

48x2 − 5

√3

72xy +

31

144y2 +

(5√

3

24− 7

6

)x

+

(5√

3

18− 31

24

)y +

157

48− 5

√3

6= 0 (1.6)

Comparing this equation with Eq. 1.5 gives A, B, and C as

A =7

48=

21

144

B = −5√

3

144(1.7)

C =31

144

4

(4,3)

ba

y’

x’

–2

0

2

4

6

8

y

–2 2 4 6 8

x

Figure 1.1: Rotated and translated ellipse

and AC − B2 > 0. By the coordinate transformation

x′ =

√3

2(x − 4) +

1

2(y − 3)

y′ = −1

2(x − 4) +

√3

2(y − 3) (1.8)

which represents a counterclockwise rotation by π/6 and a translation of theorigin to (4, 3), Eq. 1.6 can be written in the new coordinates x′ and y′ as

x′2

a2+

y′2

b2− 1 = 0 (1.9)

where a = 3 and b = 2. This is the equation for an ellipse. See Figure 1.1. Theoriginal and transformed coordinate systems are shown.

1.2.2 Parametric Cubic Curves

Parametric equations use an independent parameter u, and express points on thecurve by expressing the coordinates of the points as functions of the parameter.For example, a three-dimensional curve is expressed with three equations

x = x(u)

y = y(u) (1.10)

z = z(u)

5

A point on the three-dimensional curve is given by the row vector

p =[

x(u) y(u) z(u)]

(1.11)

This form gives more freedom for controlling the shape of a curve, compared toa nonparametric form. It is also easier to handle infinite slopes and transformingthe curve into another coordinate system.

Algebraic Form

A parametric cubic (pc) curve segment is expressed by the three polynomials

x(u) = a3xu3 + a2xu2 + a1xu + a0x

y(u) = a3yu3 + a2yu2 + a1yu + a0y (1.12)

z(u) = a3zu3 + a2zu

2 + a1zu + a0z

where u ∈ [0, 1]. The restriction of u makes the curve segment bounded, sincethe polynomials given are continuous functions in a closed bounded interval.This choice of the interval for u simplifies creation of composite curves. Acurve parametrized within another interval can always be reparametrized to theinterval u ∈ [0, 1], see page 11. The 12 coefficients in Eq. 1.12 are called algebraic

coefficients. The equation can be written more compactly in vector notation

p(u) = a3u3 + a2u

2 + a1u + a0 (1.13)

and by introducing a 1 × 4 matrix U and a 4 × 3 matrix A, where a0, a1, a2,and a3 are row vectors.

U = [ u3 u2 u 1 ] (1.14)

A =

a3

a2

a1

a0

=

a3x a3y a3z

a2x a2y a2z

a1x a1y a1z

a0x a0y a0z

(1.15)

the equation for a point on the curve can now be written

p = UA (1.16)

As an example, consider the pc curve segment specified by the matrix A, andparametrized in the interval u ∈ [0, 1]

A =

a3

a2

a1

a0

=

−1 0 00 −12 −30

12 15 303 4 0

(1.17)

The curve is plotted in Figure 1.2 on page 7.

6

0

2

4

6

8

10

z

2 4 6 8 10 12 14

y

5

10

15

x

Figure 1.2: Parametric cubic curve

Geometric Form

The algebraic form has the drawback that it is not easy to get an intuitive senseof the shape of the curve when choosing the coefficients. The pc curve can alsobe defined in terms of the coordinates and tangent vectors at the end points. Acubic curve defined in this way is called a Hermite curve. Let pu(u) denote thederivative of p(u) with respect to u

pu(u) =dp(u)

du(1.18)

By using Eq. 1.13 we get a set of four equations

p(0) = a0

p(1) = a3 + a2 + a1 + a0

pu(0) = a1 (1.19)

pu(1) = 3a3 + 2a2 + a1

which has the solution

a0 = p(0)

a1 = pu(0)

a2 = −3p(0) + 3p(1) − 2pu(0) − pu(1) (1.20)

a3 = 2p(0) − 2p(1) + pu(0) + pu(1)

These coefficients are put into Eq. 1.13, and after rearranging the terms we get

p(u) = (2u3 − 3u2 + 1)p(0) + (−2u3 + 3u2)p(1)

+ (u3 − 2u2 + u)pu(0) + (u3 − u2)pu(1) (1.21)

7

The equation can be written in the geometric form

p(u) = F1(u)p(0) + F2(u)p(1) + F3(u)pu(0) + F4(u)pu(1) (1.22)

where p(0), p(1), pu(0), and pu(1) are called the geometric coefficients. Thefunctions F1, F2, F3, and F4 are called blending functions, and are given by

F1(u) = 2u3 − 3u2 + 1

F2(u) = −2u3 + 3u2

F3(u) = u3 − 2u2 + u (1.23)

F4(u) = u3 − u2

The blending functions F1 and F2 are plotted in Figure 1.3, and F3 and F4 inFigure 1.4 on page 9. Eq. 1.22 represents a point on the curve as a weightedsum of the geometric coefficients, with the blending functions acting as weights.For u = 0, all blending functions are 0, except F1, which is 1, and we getp(0) = 1 ·p(0) as expected. As u increases, the other geometric coefficients startto contribute to the weighted sum. When u = 1, F2 = 1 and F1 = F3 = F4 = 0,and we get p(1) = 1 · p(1). Differentiating Eq. 1.22 with respect to u gives

pu(u) = F1u(u)p(0) + F2u(u)p(1) + F3u(u)pu(0) + F4u(u)pu(1) (1.24)

where F1u(u), F2u(u), F3u(u), and F4u(u) are given by

F1u(u) = 6u2 − 6u

F2u(u) = −6u2 + 6u

F3u(u) = 3u2 − 4u + 1 (1.25)

F4u(u) = 3u2 − 2u

Eq. 1.24 represents the tangent vector of a point on the curve as a weightedsum of the geometric coefficients, where the weights are given by Eq. 1.25. Foru = 0, F3u = 1, and F1u = F2u = F4u = 0, and we get pu(0) = 1 · pu(0).In the same way, for u = 1, we get F4u = 1, and F1u = F2u = F3u = 0,and pu(1) = 1 · pu(1). This is the reason for the name blending functions, thefunctions blend the contributions of the geometric coefficients. Eq. 1.22 canbe expressed as

p = FB (1.26)

where F is the 1 × 4 matrix of blending functions, and B denotes the 4 × 3matrix of geometric coefficients

F = [ F1 F2 F3 F4 ] (1.27)

B =

p(0)p(1)pu(0)pu(1)

=

x(0) y(0) z(0)x(1) y(1) z(1)xu(0) yu(0) zu(0)xu(1) yu(1) zu(1)

(1.28)

8

F2(u)F1(u)

0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

u

Figure 1.3: Blending functions F1 and F2

F3(u)

F4(u)

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8 1

u

Figure 1.4: Blending functions F3 and F4

9

The matrix of blending functions, F , can be written as a product of the Umatrix given by Eq. 1.14 and a 4 × 4 matrix M

F = UM = [ u3 u2 u 1 ]

2 −2 1 1−3 3 −2 −1

0 0 1 01 0 0 0

(1.29)

and using this expression for F in Eq. 1.26 gives

p = FB = UMB (1.30)

Comparing this expression with Eq. 1.16 gives

p = UA = UMB (1.31)

This is a matrix expression for three equations, one equation for each of x(u),y(u), and z(u). The equations hold for every u, and by choosing three lin-ear independent vectors Ui for i = 1, 2, 3, it is possible to write a formula forconverting from the geometric form to the algebraic form

A = MB (1.32)

The inverse of M exists, since the determinant of M is non-zero, (−1), and isgiven by

M−1 =

0 0 0 11 1 1 10 0 1 03 2 1 0

(1.33)

Conversion from the algebraic form to the geometric form is given by

B = M−1A (1.34)

This equation of the geometric coefficients expressed as a linear combination ofthe algebraic coefficients is the same as Eq. 1.19, as expected.

The equation expressing the tangent vector of a point on a curve, Eq. 1.24,can also be written in a compact form

pu = UMuB = [ u3 u2 u 1 ]

0 0 0 06 −6 3 3

−6 6 −4 −20 0 1 0

p(0)p(1)pu(0)pu(1)

(1.35)

The matrices U , F , M , and Mu are the same for all parametric cubic curves,parametrized by u ∈ [0, 1]. The position and shape of the curve is determinedcompletely by either the A or the B matrix, together with u ∈ [0, 1]. Asan example of a pc curve specified by the geometric form, consider the curvesegment specified by the matrix B, and parametrized in the interval u ∈ [0, 1]

B =

p(0)p(1)pu(0)pu(1)

=

3 3 013 6 512 18 612 16 16

(1.36)

The curve is plotted in Figure 1.5 on page 11.

10

012345678

z

24

68

10

y

24

68

1012

14

x

Figure 1.5: Parametric cubic curve

Reparametrization

Reparametrization refers to a change in the parametric interval of a givenparametrized curve, while maintaining the position and shape of the curve.In the previous text, the interval of the parameter for a pc curve was u ∈ [0, 1].Now, consider a curve originally parametrized in the interval u ∈ [ui, uj ], andspecified in the geometric form with coordinates p(ui), p(uj) and tangent vec-tors pu(ui), pu(uj) at the end points. Let the new parameter v be expressed asa linear function of u

v = au + b (1.37)

The linear relationship between the parameters u and v is needed in order topreserve the cubic form of the curve. The new parameter interval is v ∈ [vi, vj ],where vi and vj are given by

vi = aui + b (1.38)

vj = auj + b (1.39)

The above equations give a and b as

a =vj − vi

uj − ui(1.40)

b = vi −vj − vi

uj − uiui (1.41)

Denote the geometric coefficients of the curve parametrized by v as q(vi), q(vj),qv(vi), and qv(vj). See Figure 1.6 on page 12. A tangent vector for a point onthe curve parametrized by v is

qv =dq(v)

dv=

dq(u)

du· du

dv= qu · 1

a= qu · uj − ui

vj − vi(1.42)

11

r

r

p(ui) → q(vi)

p(uj) → q(vj)

�pu(ui) → qv(vi)

HHHjpu(uj) → qv(vj)

Figure 1.6: Reparametrization

where we have used the chain rule for differentiation, and Eq. 1.37 and Eq. 1.40.The expressions for the geometric coefficients of the curve parametrized by vbecomes

q(vi) = p(ui)

q(vj) = p(uj)

qv(vi) =uj − ui

vj − vipu(ui) (1.43)

qv(vj) =uj − ui

vj − vipu(uj)

As an example, the reparametrization from u ∈ [0, 1] to v ∈ [vi, vj ] is given by

v = vi + (vj − vi)u

q(vi) = p(0)

q(vj) = p(1) (1.44)

qv(vi) =1

vj − vipu(0)

qv(vj) =1

vj − vipu(1)

Composite Parametric Cubic Curves

Consider the following problem: two disjoint pc curves p1 and p3 are given, andwe wish to define a pc curve p2, connecting the given curves. See Figure 1.7on page 13. p1 and p3 are specified with the geometric coefficients B1 and B3

respectively.

B1 =

p1(0)p1(1)p1u(0)p1u(1)

(1.45)

12

r

p1(0)

�p1u(0)

r

p1(1)��:

p1u(1)

r

p3(0)PPPq

p3u(0)

r

p3(1)�

��>

p3u(1)

Figure 1.7: Composite curves

B3 =

p3(0)p3(1)p3u(0)p3u(1)

(1.46)

Before connecting the curve p2 in between p1 and p3, we have to decide thedegree of continuity we want. If we are only interested of connecting the curveswithout gaps, it is sufficient to state that

p2(0) = p1(1) (1.47)

p2(1) = p3(0) (1.48)

this is called C0 continuity. If we also state that the tangent vectors at theend points of p2 should be equal to the tangent vectors at p1(1) and p3(0)respectively, we get C1 continuity, that is, we have the additional requirement

p2u(0) = p1u(1) (1.49)

p2u(1) = p3u(0) (1.50)

Now, the pc curve p2 is completely specified, in summary

B2 =

p1(1)p3(0)p1u(1)p3u(0)

(1.51)

We can not in general achieve C2 continuity at a connection of two parametriccubic curves. In order to get C2 continuity, one has to use a curve with a higherdegree.

Sometimes, it is sufficient to state C0 continuity, and that the tangent vectorsat the connecting point should have the same direction, the magnitudes of the

13

tangent vectors need not to be same. This is called geometric continuity, and isdenoted G1. For the curve p2, the condition can be written

p2u(0) = k1p1u(1)

|p1u(1)| (1.52)

p2u(1) = k2p3u(0)

|p3u(0)| (1.53)

In this way, there is some freedom of varying the shape of p2 by varying k1 andk2.

1.2.3 Bezier Curves

A Bezier curve of degree n is defined by

p(u) =

n∑

i=0

piBi,n(u) , u ∈ [0, 1] (1.54)

where p0,p1, . . . ,pn are given control points, and the blending functions Bi,n(u),also called Bernstein polynomials, defined by

Bi,n(u) =

{ (ni

)(1 − u)n−iui , if 0 ≤ i ≤ n

0 , otherwise(1.55)

where the binomial coefficients(ni

)are defined by

(n

i

)=

n!

(n − i)!i!, for 0 ≤ i ≤ n (1.56)

The binomial coefficients arise from the binomial theorem, which states that

(x + y)n =

n∑

i=0

(n

i

)xn−iyi (1.57)

for any real numbers x and y, and for any natural number n.The polygon with the control points p0,p1, . . . ,pn in this order as vertices,

with an edge from pn to p0 is called the control polygon of the Bezier curve. Asan example, consider a planar cubic Bezier curve with control points

p0 = (1, 1)

p1 = (2, 3)

p2 = (4, 5)

p3 = (6, 2)

Using Eq. 1.54 with n = 3 gives

p(u) = (1 − u)3p0 + 3(1 − u)2up1 + 3(1 − u)u2p2 + u3p3 (1.58)

14

p0

p2

p3

p1

0

1

2

3

4

5

6

y

1 2 3 4 5 6 7 8

x

Figure 1.8: Planar cubic Bezier curve

and after expanding this equation and using the given control points, we canwrite the equations for each coordinate for a point on the curve as

x(u) = −u3 + 3u2 + 3u + 1 (1.59)

y(u) = −5u3 + 6u + 1 (1.60)

where u ∈ [0, 1]. The curve is shown i Figure 1.8, which also shows the controlpoints and the control polygon. The corresponding Bernstein polynomials ofdegree 3, Bi,3(u) for i = 0, . . . , 3 is plotted in Figure 1.9 on page 16.

Properties of the Bernstein Polynomials

The Bernstein polynomials Bi,n(u) have some useful properties, which will beused for proving properties of Bezier curves.

Theorem 1.1 Partition of Unity

The Bernstein polynomials of degree n sum to 1, that is

n∑

i=0

Bi,n(u) = 1 , u ∈ [0, 1] (1.61)

Proof:We can write 1 as (1 − u) + u and use the binomial theorem

1 = 1n = ((1 − u) + u)n =

n∑

i=0

(n

i

)(1 − u)n−iui =

n∑

i=0

Bi,n(u) (1.62)

which proofs the theorem.2

15

B(3,3)(u)

B(2,3)(u)B(1,3)(u)

B(0,3)(u)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

u

Figure 1.9: Bernstein polynomials of degree 3

Theorem 1.2 Positivity

The Bernstein polynomials are non-negative on the interval [0, 1]

Bi,n(u) ≥ 0 , u ∈ [0, 1] (1.63)

Proof:By writing Bi,n(u) using Eq. 1.55 and Eq. 1.56 we get

Bi,n(u) =

(n

i

)(1 − u)n−iui =

n!

(n − i)!i!(1 − u)n−iui (1.64)

The binomial coefficient(ni

)> 0, and since u ∈ [0, 1], (1 − u) ≥ 0 and

(1 − u)n−i ≥ 0 (1.65)

ui ≥ 0 (1.66)


The partition of unity and positivity properties are used later for proving theconvex hull property of a Bezier curve, and that an affine transformation of aBezier curve is also a Bezier curve. See page 23.

16

Theorem 1.3 Symmetry

The Bernstein polynomials have the following symmetry

Bn−i,n(u) = Bi,n(1 − u) , for 0 ≤ i ≤ n (1.67)

Proof:The left-hand side of Eq. 1.67 can be written

Bn−i,n(u) =n!

(n − (n − i))!(n − i)!(1 − u)n−(n−i)un−i

=n!

i!(n − i)!(1 − u)iun−i (1.68)

The right-hand side of Eq. 1.67 becomes

Bi,n(1 − u) =n!

(n − i)!i!(1 − (1 − u))n−i(1 − u)i

=n!

(n − i)!i!un−i(1 − u)i (1.69)


The positivity and symmetry properties can be seen in Figure 1.9 on page 16,for n = 3. The symmetry property implies that a symmetric control polygongives a symmetric Bezier curve.

Lemma 1.1 The following expression holds for the binomial coefficients

(n

i

)+

(n

i + 1

)=

(n + 1

i + 1

)(1.70)

Proof:(

n

i

)+

(n

i + 1

)=

n!

(n − i)!i!+

n!

(n − (i + 1))!(i + 1)!=

n!(i + 1)

(n − i)!i!(i + 1)+

(n − i)n!

(n − i)(n − i − 1)!(i + 1)!=

n!(i + 1)

(n − i)!(i + 1)!+

(n − i)n!

(n − i)!(i + 1)!=

n!i + n! + nn! − in!

(n − i)!(i + 1)!=

n!(n + 1)

(n − i)!(i + 1)!=

(n + 1)!

((n + 1) − (i + 1))!(i + 1)!=

(n + 1

i + 1

)(1.71)

2

17

Theorem 1.4 Recursion

The Bernstein polynomials of degree n can be expressed in terms of the polyno-

mials of degree n − 1

Bi,n(u) = (1 − u)Bi,n−1(u) + uBi−1,n−1(u) , for 0 ≤ i ≤ n (1.72)

where, according to the definition of the Bernstein polynomials, Eq. 1.55,

B−1,n−1(u) = 0 (1.73)

Bn,n−1(u) = 0 (1.74)

Proof:For i = 0, we get by Eq. 1.55

B0,n(u) =

(n

0

)(1 − u)nu0 = (1 − u)n (1.75)

B0,n−1(u) = (1 − u)n−1 (1.76)

By using Eq. 1.73 and the equations above, we can write

B0,n(u) = (1 − u)B0,n−1(u) + uB−1,n−1(u) (1.77)

For i = n, we get by Eq. 1.55

Bn,n(u) =

(n

n

)(1 − u)0un = un (1.78)

Bn−1,n−1(u) = un−1 (1.79)

By using Eq. 1.74 and the equations above, we can write

Bn,n(u) = (1 − u)Bn,n−1(u) + uBn−1,n−1(u) (1.80)

For 1 ≤ i ≤ n − 1 we write the right-hand side of Eq. 1.72 using Eq. 1.55, andLemma 1.1

(1 − u)Bi,n−1(u) + uBi−1,n−1(u) =(n − 1

i

)(1 − u)n−1−i+1ui +

(n − 1

i − 1

)(1 − u)n−1−(i−1)ui−1+1 =

[(n − 1

i

)+

(n − 1

i − 1

)](1 − u)n−iui =

(n

i

)(1 − u)n−iui =

Bi,n(u) (1.81)

We thus get for 0 ≤ i ≤ n

Bi,n(u) = (1 − u)Bi,n−1(u) + uBi−1,n−1(u) (1.82)

and the proof is complete.2

18

The recursion property is used later for proving the de Casteljau algorithm. Seepage 26.

Theorem 1.5 First Derivative

The first derivative of the Bernstein polynomials Bi,n(u) satisfies

d

duBi,n(u) = n(Bi−1,n−1(u) − Bi,n−1(u)) (1.83)

Proof:Differentiating Bi,n(u) by the product rule gives

d

duBi,n(u) =

n!

(n − i)!i!

d

du((1 − u)n−iui) =

n!

(n − i)!i!(−(n − i)(1 − u)n−i−1ui + (1 − u)n−iiui−1) =

−(n − i)n!

(n − i)!i!(1 − u)n−i−1ui +

n!i

(n − i)!i!(1 − u)n−iui−1 =

−n(n − 1)!

(n − i − 1)!i!(1 − u)n−1−iui +

+n(n − 1)!

((n − 1) − (i − 1))!(i − 1)!(1 − u)(n−1)−(i−1)ui−1 =

−nBi,n−1(u) + nBi−1,n−1(u) =

n(Bi−1,n−1(u) − Bi,n−1(u)) (1.84)


Convex Hull

Let X be a given set of points, where the points xi are represented as vectors.

X = {x0,x1, . . . ,xn} (1.85)

The convex hull of X, denoted by CH {X}, is defined to be the following set ofpoints

CH {X} =

{a0x0 + a1x1 + · · · + anxn

∣∣∣∣∣n∑

i=0

ai = 1, ai ≥ 0

}(1.86)

For points in a plane, one can visualize the convex hull of the set of points byplacing an elastic band around the set of points. The band is shrunk to forma convex polygon, with vertices which are a subset of the original set of points.The region bounded by the polygon is the convex hull of the points. For pointsin space, the elastic band is replaced by an elastic balloon, which is shrunkaround the points to form a convex polyhedron. The convex hull is the regionbounded by the polyhedron. See Figure 1.10 on page 20.

19

r

r

r

r r

r

r

r��

��

��

��AAAAAA

!!!!!!!!@

@@

r

r

r

r

r

r

r

��@

@@

LLLLLLLL

PPPPP

��

��

Figure 1.10: Convex hull of points in a plane and in space

Properties of Bezier Curves

Bezier curves have some properties which are useful for modeling, since it iseasy to get an intuitive sense of the shape of the curve.

Theorem 1.6 End Point Interpolation Property

A Bezier curve p(u) of degree n with control points p0,p1, . . . ,pn satisfies

p(0) = p0 (1.87)

p(1) = pn (1.88)

Proof:Using the definition of the Bezier curve, Eq. 1.54, and writing the terms in thesum, we get

p(u) =

n∑

i=0

piBi,n(u) =

n∑

i=0

pi

(n

i

)(1 − u)n−iui =

p0

(n

0

)(1 − u)n + p1

(n

1

)(1 − u)n−1u + · · ·

· · · + pn−1

(n

n − 1

)(1 − u)un−1 + pn

(n

n

)un (1.89)

For u = 0, all terms in the sum are 0 except the first, so

p(0) = p0

(n

0

)(1 − 0)n = p0 · 1 · 1 = p0 (1.90)

For u = 1, all terms are 0 except the last, and we get

p(1) = pn

(n

n

)1n = pn · 1 · 1 = pn (1.91)


20

p3p2

p1

p0,p5

p4

0

1

2

3

4

5

y

1 2 3 4 5 6 7 8

x

Figure 1.11: Closed planar Bezier curve of degree 5

The end point interpolation property can be seen in Figure 1.8 on page 15.As a consequence of this property, we can get a closed Bezier curve by settingp0 = pn. For example, consider the Bezier curve of degree 5, defined by thecontrol points

p0 = (3, 4)

p1 = (1, 2)

p2 = (2, 1)

p3 = (6, 1)

p4 = (5, 3)

p5 = (3, 4)

The coordinates for a point p(u) on the curve is given by

x(u) = 20u5 − 40u4 + 30u2 − 10u + 3 (1.92)

y(u) = −5u5 + 5u4 + 10u2 − 10u + 4 (1.93)

where u ∈ [0, 1]. The curve is shown i Figure 1.11, which also shows the controlpoints and the control polygon.

Theorem 1.7 First Derivative

The first derivative of a Bezier curve p(u) of degree n with control points

p0,p1, . . . ,pn can be written as a Bezier curve of degree n − 1

pu(u) =d

dup(u) =

n−1∑

i=0

qiBi,n−1(u) (1.94)

where the control points qi is given by

qi = n(pi+1 − pi) (1.95)

21

Proof:A Bezier curve of degree n is given by

p(u) =

n∑

i=0

piBi,n(u) (1.96)

The first derivative of p(u) is

pu(u) =

n∑

i=0

pid

duBi,n(u) (1.97)

Using Theorem 1.5 we can write

pu(u) =

n∑

i=0

pin(Bi−1,n−1(u) − Bi,n−1(u))

=n∑

i=0

pinBi−1,n−1(u) −n∑

i=0

pinBi,n−1(u) (1.98)

Since B−1,n−1(u) = 0 and Bn,n−1(u) = 0, we can remove these terms from theabove equation, which gives

pu(u) =

n∑

i=1

pinBi−1,n−1(u) −n−1∑

i=0

pinBi,n−1(u) (1.99)

Renumbering the first sum by replacing i with i + 1 gives

pu(u) =

n−1∑

i=0

pi+1nBi,n−1(u) −n−1∑

i=0

pinBi,n−1(u) (1.100)

The two sums are then written as a single sum

pu(u) =n−1∑

i=0

n(pi+1 − pi)Bi,n−1(u)

=n−1∑

i=0

qiBi,n−1(u) (1.101)

where qi = n(pi+1 − pi). This expresses a Bezier curve of degree n − 1, withcontrol points qi, and the proof is complete.2

Theorem 1.8 End Point Tangent Property

The tangent vectors at the end points of a Bezier curve p(u) of degree n with

control points p0,p1, . . . ,pn can be written

pu(0) = n(p1 − p0) (1.102)

pu(1) = n(pn − pn−1) (1.103)

22

Proof:By Theorem 1.7 we can write

pu(u) = q(u) =

n−1∑

i=0

qiBi,n−1(u) (1.104)

whereqi = n(pi+1 − pi) (1.105)

Using Theorem 1.6, where the degree of q(u) is n − 1, we get

q(0) = q0 (1.106)

q(1) = qn−1 (1.107)

Using Eq. 1.105 we can write

pu(0) = q(0) = q0 = n(p1 − p0) (1.108)

pu(1) = q(1) = qn−1 = n(pn − pn−1) (1.109)


Theorem 1.9 Convex Hull Property

Every point on a Bezier curve lies inside the convex hull of its control points.

That is

p(u) ∈ CH {p0,p1, . . . ,pn} , for u ∈ [0, 1] (1.110)

Proof:By the definition of convex hull, Eq. 1.86, it is sufficient to show that each pointp(u) on a Bezier curve can be written

p(u) = a0p0 + a1p1 + · · · + anpn (1.111)

for some constants ai ≥ 0 satisfying

n∑

i=0

ai = 1 (1.112)

Let ai = Bi,n(u). By the properties positivity and partition of unity of theBernstein polynomials, the conditions of ai are satisfied which proofs the theo-rem.2

The convex hull property of a Bezier curve can be seen in Figure 1.11 on page 21.

Theorem 1.10 Invariance under Affine Transformations

Let T be an affine transformation, for example a rotation, translation, scaling,

reflection, or a combination of these. The transformation of the Bezier curve

23

p(u) can be written as a Bezier curve where the control points are transformed

by T , that is

T

(n∑

i=0

piBi,n(u)

)=

n∑

i=0

T (pi)Bi,n(u) (1.113)

Proof:Let the affine transformation T be given by the matrix expression

T (p) = pA + b (1.114)

The transformation of a point p(u) on the curve is

T (p(u)) =

[n∑

i=0

piBi,n(u)

]A + b (1.115)

By the partition of unity property of Bi,n(u) we can write this as

T (p(u)) =

[n∑

i=0

piBi,n(u)

]A +

[n∑

i=0

Bi,n(u)

]b (1.116)

and then write as a single sum

T (p(u)) =

n∑

i=0

(piA + b)Bi,n(u) =

n∑

i=0

T (pi)Bi,n(u) (1.117)


This means that instead of transforming each point p(u) of a Bezier curve, it issufficient to transform only the control points.

Changing a Control Point

If a control point is changed, the entire curve is affected. This is called global

control. This can be seen by studying the Bernstein polynomials, which are allnon-zero for u ∈ (0, 1).

For example, consider the cubic Bezier curve in Figure 1.12 on page 25, whenchanging p1 from (2, 3) to (3, 5). The new control point p1 = (3, 5) lies on theline through p0 and the original p1. By the end point tangent property, thedirection of the curve at u = 0 is thus unchanged.

As another example, if p1 is changed to (1, 4), the direction of the curve atu = 0 is changed, since (1, 4) does not lie on the line through p0 and the originalp1. See Figure 1.13 on page 25, which shows both the old and the new curve.

Control Point with Multiplicity

If we choose k consecutive control points to be equal, we get a control pointwith multiplicity k. This has the effect of pulling the curve closer to that point.

24

new p1

p3

p2

old p1

new curve

old curve

p0

0

1

2

3

4

5

6

y

1 2 3 4 5 6 7 8

x

Figure 1.12: Planar cubic Bezier curves: p1 is changed, direction at u = 0 isnot changed

p3

p2

new curvenew p1

old p1 old curve

p0

0

1

2

3

4

5

6

y

1 2 3 4 5 6 7 8

x

Figure 1.13: Planar cubic Bezier curves: p1 is changed, direction at u = 0 ischanged

25

p2

p1

p3,p4

p0,p6

p5

0

1

2

3

4

5

y

1 2 3 4 5 6 7 8

x

Figure 1.14: Closed planar Bezier curve of degree 6 with a multiple point at(6, 1)

For example, consider the Bezier curve in Figure 1.11 on page 21. By choosinga point with multiplicity 2 at (6, 1) we get the curve shown in Figure 1.14. Sincewe have added an extra control point, the degree of the curve is increased to 6.

The de Casteljau Algorithm

The de Casteljau algorithm is a method for computing a point p(u) on a Beziercurve, for a given value of the parameter u ∈ [0, 1]. The algorithm can also beused for division of a Bezier curve into two Bezier curves, see page 29.

Theorem 1.11 The de Casteljau Algorithm

Let a Bezier curve p(u) of degree n be defined by control points p0, . . . ,pn, and

let u ∈ [0, 1] be any given parameter value. Then p(u) = pn0 where

p0i = pi (1.118)

pji = p

j−1i (1 − u) + p

j−1i+1 u (1.119)

where i and j are indices with j = 1, . . . , n and i = 0, . . . , n − j.Proof:

p(u) =

n∑

i=0

piBi,n(u) (1.120)

By the recursion property of the Bernstein polynomials, Eq. 1.72, we can write

p(u) =

n∑

i=0

pi [(1 − u)Bi,n−1(u) + uBi−1,n−1(u)]

=

n∑

i=0

pi(1 − u)Bi,n−1(u) +

n∑

i=0

piuBi−1,n−1(u) (1.121)

26

Since B−1,n−1(u) = 0 and Bn,n−1(u) = 0, we can remove these terms from theabove equation, which gives

p(u) =

n−1∑

i=0

pi(1 − u)Bi,n−1(u) +

n∑

i=1

piuBi−1,n−1(u) (1.122)

Renumbering the second sum by replacing i with i + 1 gives

p(u) =

n−1∑

i=0

pi(1 − u)Bi,n−1(u) +

n−1∑

i=0

pi+1uBi,n−1(u)

=

n−1∑

i=0

[pi(1 − u) + pi+1u]Bi,n−1(u) (1.123)

Set p1i = pi(1 − u) + pi+1u and p0

i = pi. Thus, for i = 0, . . . , n − 1 we have

p1i = p0

i (1 − u) + p0i+1u (1.124)

p(u) =

n−1∑

i=0

p1i Bi,n−1(u) (1.125)

This equation expresses p(u) as a Bezier curve of degree n−1 with control pointsp1

0, . . . ,p1n−1. Applying a similar argument to Eq. 1.125 gives for i = 0, . . . , n−2

p2i = p1

i (1 − u) + p1i+1u (1.126)

p(u) =n−2∑

i=0

p2i Bi,n−2(u) (1.127)

and in general, for i = 0, . . . , n − j

pji = p

j−1i (1 − u) + p

j−1i+1 u (1.128)

p(u) =

n−j∑

i=0

pjiBi,n−j(u) (1.129)

In particular, j = n gives

p(u) =0∑

i=0

pni Bi,0(u) = pn

0B0,0(u) = pn0 (1.130)


The de Casteljau algorithm for a cubic Bezier curve can be depicted with thefollowing diagram

27

p00 p0

1 p02 p0

3

↓ (1 − u) ւ u ↓ (1 − u) ւ u ↓ (1 − u) ւ u

j = 1 p10 p1

1 p12

↓ (1 − u) ւ u ↓ (1 − u) ւ u

j = 2 p20 p2

1

↓ (1 − u) ւ u

j = 3 p30

For example, p10 is computed as p1

0 = p00(1 − u) + p0

1u, for a given value of theparameter u. As a numeric example, consider the cubic Bezier curve given inFigure 1.8 on page 15, with the control points

p0 = (1, 1)

p1 = (2, 3)

p2 = (4, 5)

p3 = (6, 2)

To compute the point on the curve p(u) at u = 0.5 we get

(1, 1) (2, 3) (4, 5) (6, 2)

↓ 0.5 ւ 0.5 ↓ 0.5 ւ 0.5 ↓ 0.5 ւ 0.5

(1.5, 2) (3, 4) (5, 3.5)

↓ 0.5 ւ 0.5 ↓ 0.5 ւ 0.5

(2.25, 3) (4, 3.75)

↓ 0.5 ւ 0.5

(3.125, 3.375)

By computing p(u) for u = 0.5 using Eq. 1.59 and Eq. 1.60 for x(u) and y(u)respectively, we also get (3.125, 3.375), as expected.

28

Subdivision of a Bezier Curve

A Bezier curve p(u) is in general defined on the interval u ∈ [0, 1] and given by

p(u) =

n∑

i=0

piBi,n(u) (1.131)

Suppose that the curve is cut at u = α to give two curve segments, denotedpleft (u) and pright (u) defined on the intervals u ∈ [0, α] and u ∈ [α, 1] respec-tively. Theorem 1.12 on page 32 gives expressions for the two curve segments.The following two lemmas are used in the proof of the theorem.

Lemma 1.2 The pjk computed for u = α in the de Casteljau algorithm satisfy

pjk =

j∑

i=0

pi+kBi,j(α) (1.132)

Proof:The lemma is proved by induction over j. In the de Casteljau algorithm p

jk are

given byp

jk = p

j−1k (1 − α) + p

j−1k+1α (1.133)

for j = 1, . . . , n and k = 0, . . . , n − j. p0k = pk for k = 0, . . . , n.

For j = 1, the left-hand side of Eq. 1.132 is

p1k = p0

k(1 − α) + p0k+1α = pk(1 − α) + pk+1α (1.134)

and the right-hand side is

1∑

i=0

pi+kBi,1(α) = pkB0,1(α) + pk+1B1,1(α)

= pk

(1

0

)(1 − α)1α0 + pk+1

(1

1

)(1 − α)0α1

= pk(1 − α) + pk+1α (1.135)

That is, Eq. 1.132 is true for j = 1.The induction hypothesis is that Eq. 1.132 is true for j = p < n

ppk =

p∑

i=0

pi+kBi,p(α) (1.136)

We want to show that it is also true for j = p+1. By the de Casteljau algorithmwith j = p + 1 we get

pp+1k = p

pk(1 − α) + p

pk+1α (1.137)

Using the induction hypothesis, Eq. 1.136, we can write the above equation as

pp+1k =

p∑

i=0

pi+kBi,p(α)(1 − α) +

p∑

i=0

pi+k+1Bi,p(α)α (1.138)

29

The second sum is renumbered by replacing i + 1 by i

pp+1k =

p∑

i=0


p+1∑

i=1

pi+kBi−1,p(α)α (1.139)

Since Bp+1,p(α) = 0 and B−1,p(α) = 0, we can write the sums as

pp+1k =

p+1∑

i=0


p+1∑

i=0

pi+kBi−1,p(α)α

=

p+1∑

i=0

pi+k(Bi,p(α)(1 − α) + Bi−1,p(α)α) (1.140)

Using the recursion property of the Bernstein polynomials, the equation can bewritten

pp+1k =

p+1∑

i=0

pi+kBi,p+1(α) (1.141)

By the induction axiom, it follows that

pjk =

j∑

i=0

pi+kBi,j(α) (1.142)

for j = 1, . . . , n and the proof is complete.2

Lemma 1.3 The following property holds for the Bernstein polynomials

Bi,n(αu) =

n∑

j=0

Bi,j(α)Bj,n(u) , 0 ≤ i ≤ n (1.143)

where u ∈ [0, 1] and α ∈ [0, 1].Proof:The lemma is proved by induction over n.For n = 0 and 0 ≤ i ≤ n the left-hand side of Eq. 1.143 is

B0,0(αu) =

(0

0

)(1 − αu)0(αu)0 = 1 (1.144)

and the right-hand side is

0∑

j=0

B0,j(α)Bj,0(u) = B0,0(α)B0,0(u) = 1 · 1 = 1 (1.145)

That is, Eq. 1.143 is true for n = 0.The induction hypothesis is that Eq. 1.143 is true for n = p

Bi,p(αu) =

p∑

j=0

Bi,j(α)Bj,p(u) , 0 ≤ i ≤ p (1.146)

30

We want to show that it is also true for n = p+1. Using the recursion propertyof Bernstein polynomials we can write

Bi,p+1(αu) = (1 − αu)Bi,p(αu) + αuBi−1,p(αu) (1.147)

Using the induction hypothesis, Eq. 1.146, we can write the above equation as

Bi,p+1(αu) = (1 − αu)

p∑

j=0

Bi,j(α)Bj,p(u) + αu

p∑

j=0

Bi−1,j(α)Bj,p(u) (1.148)


Bi,p+1(αu) =

p∑

j=0

[(1 − αu)Bi,j(α) + αuBi−1,j(α)] Bj,p(u) (1.149)

Adding −uBi,j(α) + uBi,j(α) = 0 into the sum gives

Bi,p+1(αu) =

p∑

j=0

[Bi,j(α) − uBi,j(α)

+ uBi,j(α) − αuBi,j(α) + αuBi−1,j(α)] Bj,p(u) (1.150)

Bi,p+1(αu) =

p∑

j=0

[Bi,j(α)(1 − u) + u((1 − α)Bi,j(α) + αBi−1,j(α))] Bj,p(u)

(1.151)Using the recursion property of the Bernstein polynomials gives

Bi,p+1(αu) =

p∑

j=0

[Bi,j(α)(1 − u) + uBi,j+1(α)] Bj,p(u) (1.152)

Writing the sum as two sums gives

Bi,p+1(αu) =

p∑

j=0

Bi,j(α)(1 − u)Bj,p(u) +

p∑

j=0

Bi,j+1(α)uBj,p(u) (1.153)

Renumbering the second sum by replacing j by j − 1 gives

Bi,p+1(αu) =

p∑

j=0

Bi,j(α)(1 − u)Bj,p(u) +

p+1∑

j=1

Bi,j(α)uBj−1,p(u) (1.154)

Since Bp+1,p(u) = 0 and B−1,p(u) = 0 we can write the sums as

Bi,p+1(αu) =

p+1∑

j=0

Bi,j(α)(1 − u)Bj,p(u) +

p+1∑

j=0

Bi,j(α)uBj−1,p(u) (1.155)


Bi,p+1(αu) =

p+1∑

j=0

Bi,j(α) [(1 − u)Bj,p(u) + uBj−1,p(u)] (1.156)

31

Using the recursion property of the Bernstein polynomials gives

Bi,p+1(αu) =

p+1∑

j=0

Bi,j(α)Bj,p+1(u) (1.157)

By the induction axiom, it follows that

Bi,n(αu) =n∑

j=0

Bi,j(α)Bj,n(u) (1.158)


Theorem 1.12 Subdivision of a Bezier Curve

A general Bezier curve of degree n, with control points p0, . . . ,pn

p(u) =n∑

i=0

piBi,n(u) (1.159)

which is cut at u = α gives two Bezier curves pleft (u) and pright (u), both defined

on the interval u ∈ [0, 1]

pleft (u) =

n∑

j=0

pj0Bj,n(u) (1.160)

pright (u) =

n∑

i=0

pn−ii Bi,n(u) (1.161)

where the pji are the points computed for u = α in the de Casteljau algorithm.

Proof:Suppose p(u) is divided at u = α. Define pleft(u) by

pleft(u) =

n∑

i=0

piBi,n(u) , u ∈ [0, α] (1.162)

The curve can be reparametrized as

pleft(u) =

n∑

i=0

piBi,n(αu) , u ∈ [0, 1] (1.163)

By Lemma 1.3, Bi,n(αu) is

Bi,n(αu) =

n∑

j=0

Bi,j(α)Bj,n(u) (1.164)

Using this expression in the equation for pleft(u) gives

pleft(u) =

n∑

i=0

pi

n∑

j=0

Bi,j(α)Bj,n(u) (1.165)

32

Changing the order of the sums gives

pleft(u) =n∑

j=0

n∑

i=0

piBi,j(α)Bj,n(u) (1.166)

When i > j, Bi,j(α) = 0, so we can write

pleft(u) =

n∑

j=0

j∑

i=0

piBi,j(α)Bj,n(u) (1.167)

Lemma 1.2 states that

pjk =

j∑

i=0

pi+kBi,j(α) (1.168)

Using this expression with k = 0 in the equation for pleft(u) gives

pleft(u) =

n∑

j=0

pj0Bj,n(u) (1.169)

which proofs the theorem for pleft(u).Define pright(u) by

pright(u) =

n∑

i=0

piBi,n(u) , u ∈ [α, 1] (1.170)

We can reparametrize by substituting u for 1 − u

pright(u) =n∑

i=0

piBi,n(1 − u) , u ∈ [0, 1 − α] (1.171)

By the symmetry property of the Bernstein polynomials we can write

pright(u) =

n∑

i=0

piBn−i,n(u) , u ∈ [0, 1 − α] (1.172)

Renumbering the sum by replacing i by n − i gives

pright(u) =

n∑

i=0

pn−iBi,n(u) , u ∈ [0, 1 − α] (1.173)

We reparametrize for u ∈ [0, 1]

pright(u) =

n∑

i=0

pn−iBi,n((1 − α)u) , u ∈ [0, 1] (1.174)

By Lemma 1.3

Bi,n((1 − α)u) =

n∑

j=0

Bi,j(1 − α)Bj,n(u) (1.175)

33

Using this expression in the equation for pright(u) gives

pright(u) =

n∑

i=0

n∑

j=0

pn−iBi,j(1 − α)Bj,n(u) , u ∈ [0, 1] (1.176)

We reparametrize by substituting u for 1 − u

pright(u) =n∑

i=0

n∑

j=0

pn−iBi,j(1 − α)Bj,n(1 − u) , u ∈ [0, 1] (1.177)


pright(u) =

n∑

i=0

n∑

j=0

pn−iBi,j(1 − α)Bn−j,n(u) , u ∈ [0, 1] (1.178)

We renumber the inner sum by replacing j by n − j

pright(u) =n∑

i=0

n∑

j=0

pn−iBi,n−j(1 − α)Bj,n(u) , u ∈ [0, 1] (1.179)

Changing the order of the sums gives

pright(u) =

n∑

j=0

n∑

i=0

pn−iBi,n−j(1 − α)Bj,n(u) , u ∈ [0, 1] (1.180)

When i > n − j, Bi,n−j(1 − α) = 0, so we can write

pright(u) =

n∑

j=0

n−j∑

i=0

pn−iBi,n−j(1 − α)Bj,n(u) , u ∈ [0, 1] (1.181)


pright(u) =

n∑

j=0

n−j∑

i=0

pn−iBn−j−i,n−j(α)Bj,n(u) , u ∈ [0, 1] (1.182)

We renumber the inner sum by replacing i by n − j − i

pright(u) =

n∑

j=0

n−j∑

i=0

pi+jBi,n−j(α)Bj,n(u) , u ∈ [0, 1] (1.183)

By Lemma 1.2

pjk =

j∑

i=0

pi+kBi,j(α) (1.184)

With j for k, and n − j for j we can write this as

pn−jj =

n−j∑

i=0

pi+jBi,n−j(α) (1.185)

34

Using this expression in the equation for pright(u) gives

pright(u) =

n∑

j=0

pn−jj Bj,n(u) , u ∈ [0, 1] (1.186)

which proofs the theorem for pright(u), and the proof is complete.2

As an example of subdivision of a cubic Bezier curve, consider the the Beziercurve defined by the control points

p0 = (1, 1)

p1 = (2, 3)

p2 = (4, 5)

p3 = (6, 2)

These control points were also used in Figure 1.8 on page 15. The curve isdivided at u = 0.5. The two curves, pleft(u) and pright (u) are defined by thepoints computed in the de Casteljau algorithm for u = 0.5. See the diagram ofthe example on page 28. pleft (u) is defined by the control points qi = pi

0, fori = 0, . . . , 3

q0 = p00 = (1, 1)

q1 = p10 = (1.5, 2)

q2 = p20 = (2.25, 3)

q3 = p30 = (3.125, 3.375)

and pright (u) is defined by the control points ri = p3−ii , for i = 0, . . . , 3

r0 = p30 = (3.125, 3.375)

r1 = p21 = (4, 3.75)

r2 = p12 = (5, 3.5)

r3 = p03 = (6, 2)

The two curves are shown in Figure 1.15 on page 36. The original controlpolygon and the control polygons for pleft (u) and pright (u) are also shown.Compare this with Figure 1.8 on page 15.

1.2.4 Rational Bezier Curves

A rational Bezier curve of degree n is defined by

p(u) =

n∑

i=0

piRi,n(u) , u ∈ [0, 1] (1.187)

35

p1

p3

p2

p0

0

1

2

3

4

5

6

y

1 2 3 4 5 6 7 8

x

Figure 1.15: Subdivided cubic Bezier curve

where p0,p1, . . . ,pn are given control points, and the blending functions Ri,n(u)are defined by

Ri,n(u) =

wiBi,n(u)∑n

j=0wjBj,n(u)

, if wi 6= 0

Bi,n(u)∑n

j=0wjBj,n(u)

, if wi = 0(1.188)

where wi is a scalar weight for control point pi. At least one of the weightsw0, w1, . . . , wn is non-zero. If we choose all the weights to be equal and non-zero, that is, w = w0 = w1 = · · · = wn, we can factor out w from the sum inthe denominator of the first expression in Eq. 1.188, and then use the partitionof unity property of Bi,n(u) to get

Ri,n(u) =wBi,n(u)∑n

j=0 wBj,n(u)=

wBi,n(u)

w∑n

j=0 Bj,n(u)= Bi,n(u) (1.189)

that is, by setting all weights wi equal and non-zero, we get the Bezier curvesdescribed in Section 1.2.3 (often called integral Bezier curves), as special casesof rational Bezier curves.

As an example of a planar quadratic rational Bezier curve, consider the curvedefined by the control points

p0 = (1, 0)

p1 = (1, 1)

p2 = (0, 1)

and by the weights

w0 = 1

36

p1p2

p0

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

y

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1

x

Figure 1.16: Planar quadratic rational Bezier curve

w1 = 1

w2 = 2

Using Eq. 1.187 and Eq. 1.188 we can write the expression for a point of thecurve as

p(u) =w0(1 − u)2p0 + 2w1(1 − u)up1 + w2u

2p2

w0(1 − u)2 + 2w1(1 − u)u + w2u2(1.190)

By using the given control points and weights, and simplifying, we can writethe expressions for the coordinates of the point on the curve as

x(u) =1 − u2

1 + u2(1.191)

y(u) =2u

1 + u2(1.192)

This is a parametrization of a unit circle, and for u ∈ [0, 1], we get the part ofthe circle which lies in the first quadrant. See Figure 1.16.

Properties of Rational Bezier Curves

Rational Bezier curves have similar properties as integral Bezier curves.Theorems 1.13–1.16 can be proved by first proving that Ri,n(u) satisfy the posi-tivity property and the partition of unity property, provided that all the weightssatisfy wi > 0.

37


A rational Bezier curve p(u) of degree n with control points p0,p1, . . . ,pn, and

weights w0, w1, . . . , wn, where w0 6= 0 and wn 6= 0, satisfies

p(0) = p0 (1.193)

p(1) = pn (1.194)

Theorem 1.14 End Point Tangent Property

The tangent vectors at the end points of a rational Bezier curve p(u) of degree

n with control points p0,p1, . . . ,pn, and weights w0, w1, . . . , wn, where w0 6= 0and wn 6= 0, can be written

pu(0) = nw1

w0(p1 − p0) (1.195)

pu(1) = nwn−1

wn(pn − pn−1) (1.196)


Every point on a rational Bezier curve lies inside the convex hull of its control

points, provided that all weights wi > 0. That is

p(u) ∈ CH {p0,p1, . . . ,pn} , for u ∈ [0, 1] (1.197)



reflection, or a combination of these. The transformation of the rational Bezier

curve p(u) can be written as a rational Bezier curve where the control points

are transformed by T , that is

T

(n∑

i=0

piRi,n(u)

)=

n∑

i=0

T (pi)Ri,n(u) (1.198)

This means that instead of transforming each point p(u) of a rational Beziercurve, it is sufficient to transform only the control points.

1.2.5 B-Spline Curves

A B-spline curve of degree d is a piecewise polynomial curve defined by

p(u) =

n∑

i=0

piNi,d(u) , u ∈ [a, b] (1.199)

where p0,p1, . . . ,pn are given control points. The B-spline basis functions ofdegree d, Ni,d(u), for i = 0, 1, . . . , n, defined by the knot sequence u0, u1, . . . , um,are defined recursively as

Ni,0(u) =

{1 , if u ∈ [ui, ui+1)0 , otherwise

(1.200)

Ni,d(u) =u − ui

ui+d − uiNi,d−1(u) +

ui+d+1 − u

ui+d+1 − ui+1Ni+1,d−1(u) (1.201)

38

The knots in the knot sequence satisfy ui ≤ ui+1, for i = 0, . . . , m − 1, andud = a, um−d = b. The knots u0, u1, . . . , ud and um−d, um−d+1, . . . , um arecalled end knots, and the knots ud+1, ud+2, . . . , um−d−1 are called interior knots.The knot sequence can contain repeated knots. The number of times a knotvalue occurs is called the multiplicity of the knot. We can define a new sequencev0, v1, . . . , vr, called the breakpoints, satisfying v0 < v1 < · · · < vr and consistingof the distinct values of the interior knots. In the equations defining the B-splinebasis functions, Eq. 1.200 and Eq. 1.201, we will get a division of the form 0/0for some i, if we use repeated knot values. When this occurs, we replace thisdivision by 0.

As an example of a quadratic planar B-spline curve, consider the curvedefined by the knot sequence

u0 = 2

u1 = 4

u2 = 5

u3 = 7

u4 = 8

u5 = 10

u6 = 11

and by the control points

p0 = (1, 2)

p1 = (3, 5)

p2 = (6, 2)

p3 = (9, 4)

The B-spline curve is shown in Figure 1.17 on page 40, together with the fourcontrol points. The points on the curve corresponding to the end knots andthe single breakpoint are also shown. Note that this B-spline curve does notinterpolate the first and last control points. The curve is defined on the interval[u2, u4] = [5, 8] and consists of two curve segments. We can write the curve asthe piecewise polynomial

p(u) =

16 (7 − u)2p0 + 1

3 (−u2 + 12u − 34)p1 + 16 (u − 5)2p2 , 5 ≤ u < 7

13 (8 − u)2p1 + 2

3 (−u2 + 15u − 55)p2 + 13 (u − 7)2p3 , 7 ≤ u ≤ 8

As we have seen before, a Bezier curve of degree n has exactly n + 1 controlpoints. A B-spline curve of degree d can have any number of control points,provided that we specify a sufficient number of knots. In this way, we canget more freedom when defining a B-spline curve, by increasing the number ofcontrol points and keep the degree fixed. As can be seen from Eq. 1.201, eachB-spline basis function Ni,d(u) is defined by the d+2 knots ui, ui+1, . . . , ui+d+1.If we have specified n + 1 control points (p0,p1, . . . ,pn) we need for N0,d(u)

39

p3

p2

p1

p0

0

1

2

3

4

5

6

y

2 4 6 8 10

x

Figure 1.17: Quadratic planar B-spline curve

the knots u0, u1, . . . , ud+1. For Nn,d(u) we need the knots un, un+1, . . . , un+d+1.Thus, in total, we need the knots u0, u1, . . . , un+d+1, that is, n + d + 2 knots.Since the number of knots is m + 1 we get the identity

m = n + d + 1 (1.202)

Properties of the B-Spline Basis Functions

The B-spline basis functions Ni,d(u) satisfy some properties, which in turn giveuseful properties of the B-spline curves. For the proofs of Theorems 1.17–1.21,see [3].

Theorem 1.17 Positivity

Ni,d(u) > 0 , for u ∈ (ui, ui+d+1) (1.203)

Theorem 1.18 Local Support

Ni,d(u) = 0 , for u 6∈ (ui, ui+d+1) (1.204)

Theorem 1.19 Piecewise Polynomial

Ni,d(u) are piecewise polynomial functions of degree d.

Theorem 1.20 Partition of Unity

r∑

j=r−d

Nj,d(u) = 1 , for u ∈ [ur, ur+1) (1.205)

Theorem 1.21 Continuity

If the interior knot ui has multiplicity pi, then Ni,d(u) is Cd−pi at u = ui.

Ni,d(u) is C∞ elsewhere.

40

Properties of B-Spline Curves

B-spline curves have some properties, which make them useful for modeling.For the proofs of Theorems 1.22–1.25, see [3].

Theorem 1.22 Local Control Property

Let a B-spline curve of degree d, defined by the knot sequence u0, u1, . . . , um,

and with the control points p0,p1, . . . ,pn, be given by

p(u) =

n∑

i=0

piNi,d(u) (1.206)

Then each segment of the curve is determined by d + 1 control points.

If u ∈ [ur, ur+1) where d ≤ r ≤ m − d − 1, then

p(u) =

r∑

i=r−d

piNi,d(u) (1.207)

That is, to evaluate p(u) it is sufficient to evaluate the d + 1 B-spline basis

functions Nr−d,d(u), . . . , Nr,d(u), instead of all n + 1 basis functions.

Each of the control points can affect at most d + 1 segments. For example,the control point p0 affects the first segment only, and p1 affects the first twosegment of the curve. Compare this with a Bezier curve, where the entire curveis affected by each of its control points, which is called global control.


Let a B-spline curve of degree d, defined by the knot sequence u0, u1, . . . , um,

and with the control points p0,p1, . . . ,pn, be given by

p(u) =

n∑

i=0

piNi,d(u) (1.208)


p(u) ∈ CH {pr−d,pr−d+1, . . . ,pr} (1.209)

That is, each segment of the curve lies in the convex hull of the control points

which determine the segment.

As an example of the properties local control and convex hull, consider theplanar cubic B-spline curve defined by the knot sequence

ui =

0 , for i = 0, 1, 2, 3(i − 3) , for i = 4, 5, 6, 7

5 , for i = 8, 9, 10, 11(1.210)


p0 = (2, 2)

41

new curve

old curve

new p2

p7

p6

p5

p4p3

old p2p1

p0

0

2

4

6

8

10

y

2 4 6 8 10 12 14

x

Figure 1.18: Cubic planar B-Spline curves, p2 is changed

p1 = (3, 6)

p2 = (6, 5)

p3 = (7, 1)

p4 = (9, 2)

p5 = (11, 6)

p6 = (12, 2)

p7 = (14, 4)

The B-spline curve is shown in Figure 1.18, together with the eight controlpoints. The effect of changing a control point is also shown. p2 is changedto (7, 8), which gives a new curve. Both curves consist of five segments. Thepoints on the curves corresponding to the breakpoints define the boundaries ofthe segments. Each segment is determined by d + 1 control points, where thedegree, d, of the curves is 3. For example, the first segment is determined byp0,p1,p2, and p3, and the second by p1,p2,p3, and p4. This means that thechange of control point p2 affects the first three segments, while the last twosegments are unaffected. Figure 1.18 also shows the convex hull property for thelast segment, that is, the last segment lies in the convex hull of control pointsp4,p5,p6, and p7.


If pi is the multiplicity of the breakpoint vi, then p(u) is Cd−pi at u = vi, and

C∞ elsewhere.



reflection, or a combination of these. The transformation of the B-spline curve

p(u) can be written as a B-spline curve where the control points are transformed

42

by T , that is

T

(n∑

i=0

piNi,d(u)

)=

n∑

i=0

T (pi)Ni,d(u) (1.211)

This means that instead of transforming each point p(u) of a B-spline curve, itis sufficient to transform only the control points.

1.2.6 B-Spline Curve Types

By choosing the knot sequence and the control points in certain ways, we canget various types of B-spline curves.

Open B-Spline Curves

An open B-spline curve is a B-spline curve where the end knots satisfy

u0 = u1 = · · · = ud

um−d = um−d+1 = · · · = um

It can be shown that for this choice of the knot sequence, we get a B-splinecurve which interpolates the end control points, that is

p(ud) = p0 (1.212)

p(um−d) = pn (1.213)

The B-spline curves in Figure 1.18 on page 42 are open B-spline curves. Notethat the end control points are interpolated.

Uniform B-Spline Curves

A B-spline curve is said to be uniform whenever its knots are equally spaced,and non-uniform otherwise.

Open Uniform B-Spline Curves

An open uniform B-spline curve is an open B-spline curve where the interiorknots are equally spaced. The B-spline curves in Figure 1.18 on page 42 areopen uniform B-spline curves.

Periodic B-Spline Curves

A periodic B-spline curve of degree d with n + 1 control points is a B-splinecurve where the first n + 1 knots, u0 ≤ u1 ≤ · · · ≤ un, are chosen arbitrarily.The remaining d + 1 knots are given by

un+i = un+i−1 + (ui − ui−1) , for i = 1, . . . , d + 1 (1.214)

The knot sequence of this form is called a periodic knot sequence. A uniformB-spline curve is a special case of a periodic B-spline curve.

43

Closed Periodic B-Spline Curves

A closed periodic B-spline curve of degree d with n+1 control points p0, . . . ,pn

is obtained by choosing a periodic knot sequence with n + 2d + 2 knots, andthen introducing d additional control points given by

pn+i = pi−1 , for i = 1, 2, . . . , d (1.215)

As an example of a closed periodic B-spline curve, consider the planar cubiccurve defined by the periodic (and uniform) knot sequence

ui = i , for i = 0, 1, . . . , 12 (1.216)


p0 = (1, 2)

p1 = (2, 4)

p2 = (4, 5)

p3 = (5, 4)

p4 = (5, 2)

p5 = (4, 1)

We add d = 3 control points, p6 = p0, p7 = p1, and p8 = p2. The curve isshown in Figure 1.19 on page 45, together with the original control points. Notethat the curve is closed, and does not interpolate any of the control points.

1.2.7 NURBS Curves

The B-spline curves described in Section 1.2.5 (often called integral B-splinecurves) can be generalized to get NURBS curves. NURBS is an abbreviationfor Non-Uniform Rational B-Spline. A NURBS curve of degree d is a piecewiserational curve defined by

p(u) =n∑

i=0

piRi,d(u) , u ∈ [a, b] (1.217)

where p0,p1, . . . ,pn are given control points. The rational B-spline basis func-tions Ri,d(u) are defined by

Ri,d(u) =

wiNi,d(u)∑n

j=0wjNj,d(u)

, if wi 6= 0

Ni,d(u)∑n

j=0wjNj,d(u)

, if wi = 0(1.218)

where Ni,d(u) are the basis functions for B-splines, defined by Eq. 1.200 andEq. 1.201, and wi is a scalar weight for control point pi. At least one of theweights w0, w1, . . . , wn is non-zero. As for integral B-spline curves, we also needa knot sequence with m + 1 knots (u0, u1, . . . , um). If all weights are chosen

44

p5

p4

p3

p2

p1

p0

0

1

2

3

4

5

6

y

1 2 3 4 5 6

x

Figure 1.19: Closed periodic cubic planar B-spline curve

equal and non-zero, we get integral B-spline curves as special cases of NURBScurves.

As an example of a quadratic NURBS curve, consider the curve defined bythe control points

p0 = (1, 0)

p1 = (1, 1)

p2 = (−1, 1)

p3 = (−1, 0)

p4 = (−1,−1)

p5 = (1,−1)

p6 = (1, 0)

and by the weights

w0, . . . , w6 = 1,1

2,1

2, 1,

1

2,1

2, 1

and by the knot sequence

u0, . . . , u9 = 0, 0, 0,1

4,1

2,1

2,3

4, 1, 1, 1

The NURBS curve is defined on the interval [u2, u7] = [0, 1]. As shown inFigure 1.20 on page 46, the curve is the unit circle. This NURBS curve is usefulwhen creating surfaces of revolution, described in Section 1.3.4.

45

p6

p5p4

p3

p2 p1

p0

–1

–0.5

0

0.5

1

y

–1 –0.5 0.5 1

x

Figure 1.20: Unit circle as a quadratic planar NURBS curve

Properties of NURBS Curves

Theorems 1.26–1.29 state some useful properties of NURBS curves.


Let a NURBS curve of degree d, defined by the knot sequence u0, u1, . . . , um,

and with the control points p0,p1, . . . ,pn, with the corresponding weights wi, be

given by

p(u) =n∑

i=0

piRi,d(u) (1.219)

Then each segment of the curve is determined by d + 1 control points.


p(u) =

r∑

i=r−d

piRi,d(u) (1.220)

That is, to evaluate p(u) it is sufficient to evaluate the d + 1 rational B-spline

basis functions Rr−d,d(u), . . . , Rr,d(u), instead of all n + 1 basis functions.


Let a NURBS curve of degree d, defined by the knot sequence u0, u1, . . . , um,

and with the control points p0,p1, . . . ,pn, with the corresponding weights wi, be

given by

p(u) =

n∑

i=0

piRi,d(u) (1.221)

46


p(u) ∈ CH {pr−d,pr−d+1, . . . ,pr} (1.222)

That is, each segment of the curve lies in the convex hull of the control points



If pi is the multiplicity of the breakpoint vi, then p(u) is Cd−pi at u = vi, and

C∞ elsewhere.



reflection, or a combination of these. The transformation of the NURBS curve

p(u) can be written as a NURBS curve where the control points are transformed

by T , that is

T

(n∑

i=0

piRi,d(u)

)=

n∑

i=0

T (pi)Ri,d(u) (1.223)

This means that instead of transforming each point p(u) of a NURBS curve, itis sufficient to transform only the control points.

1.2.8 Linear Approximations of Curves

We can approximate a curve by piecewise linear segments. For example, supposea Bezier curve of degree n is given by

p(u) =n∑

i=0

piBi,n(u) , u ∈ [0, 1] (1.224)

We can choose N + 1 values of the parameter u as

ui =i

N, i = 0, 1, . . . , N (1.225)

and evaluate p(u) at u = ui. We can then connect the points p(ui) and p(ui+1)for i = 0, 1, . . . , N − 1 by line segments. For N + 1 points on the curve we getN line segments, and each line segment Li can be parametrized as

Li(t) = p(ui)(1 − t) + p(ui+1)t , t ∈ [0, 1] (1.226)

As an example, consider the cubic Bezier curve in Figure 1.8 on page 15. If wechoose N = 5, we get ui as

ui = 0,1

5,2

5, . . . , 1 (1.227)

Figure 1.21 on page 48 shows the original curve and the approximation of thecurve by 5 piecewise linear segments. In order to get a good approximation ofthe curve, we have to choose N sufficiently large.

47

0

1

2

3

4

5

6

y

1 2 3 4 5 6 7 8

x

Figure 1.21: Linear approximation of a cubic Bezier curve

We could also use a non-uniform spacing between the parameter values ui.For example, the curvature of the curve could be used as determining the spac-ing in such a way that when the curvature is small, we use a larger distancebetween successive values of ui, and when the curvature is large, we use a smallerdistance.

Linear approximation of a curve is often used when the curve is rendered,that is, plotted on a computer screen, for example. Rendering of a Bezier curveis described in more detail in Section 3.2.3.

1.3 Representation of 3D Surfaces

This section shows how surfaces in 3 dimensions can be expressed by parametricequations. Common special cases, as parametric bicubic surfaces, Bezier sur-faces, and B-spline surfaces are described. Approximating a surface by a meshof polygons will also be discussed.

1.3.1 Explicit and Implicit Equations

For a point (x, y, z) on a surface, we can express the z coordinate as a real-valuedfunction of x and y as

z = f(x, y) (1.228)

where a ≤ x ≤ b and c ≤ y ≤ d. This formulation has the drawback thata surface with multiple values of z for the same value of the pair of x and ycoordinates, has to be broken into multiple surfaces, such that for each surfacewe get a single value for z for each pair of the x and y coordinates. For example,

48

a sphere with radius r centered at the origin can be divided into two semi spheresS1 and S2 as:

S1 : z =√

r2 − (x2 + y2) , x2 + y2 ≤ r2

S2 : z = −√

r2 − (x2 + y2) , x2 + y2 ≤ r2 (1.229)

Note that we need the constraint x2 + y2 ≤ r2.As another example, the general second-degree implicit equation

Ax2 + By2 + Cz2 + 2Dxy + 2Eyz + 2Fxz + Px + Qy + Rz + T = 0 (1.230)

can represent planes, cones, cylinders, ellipsoids, hyperboloids, and paraboloids,depending on the coefficients. For the classification, see for example [5]. As anexample, consider the equation

7

48x2 +

31

144y2 +

1

4z2 − 5

√3

72xy +

(5√

3

24− 7

6

)x

+

(5√

3

18− 31

24

)y − z +

205

48− 5

√3

6= 0 (1.231)

By the coordinate transformation

x′ =

√3

2(x − 4) +

1

2(y − 3)

y′ = −1

2(x − 4) +

√3

2(y − 3) (1.232)

z′ = z − 2

which represents a counterclockwise rotation by π/6 around the z-axis and atranslation of the origin to (4, 3, 2), Eq. 1.231 can be written in the new coordi-nates x′, y′, and z′ as

x′2

a2+

y′2

b2+

z′2

c2− 1 = 0 (1.233)

where a = 3, b = 2, and c = 2. This is the equation for an ellipsoid. SeeFigure 1.22 on page 50.

1.3.2 Parametric Bicubic Surfaces

The coordinates of a point on a surface can be expressed by three parametricfunctions, one for each of the coordinates x, y, and z. Each function takes twoparameters, u and w.

x = x(u, w)

y = y(u, w) (1.234)

z = z(u, w)

49

–1

1

2

3

4

5

z

24

68

y–2

24

68

x

Figure 1.22: Rotated and translated ellipsoid

A point on the surface is given by the row vector

p =[

x(u, w) y(u, w) z(u, w)]

(1.235)

A patch is a curve-bounded surface where the functions are continuous and theparameters u and w are constrained. Usually, u and w are constrained to theinterval [0, 1]. If one of the parameters, for example w, is fixed, we get a curvelying in the patch when we vary u. By repeating the process with differentvalues of w, and then doing the same for u, we get a net of two one-parameterfamilies of curves, such that one curve of each family passes through a point onthe patch.

Algebraic Form

A bicubic patch is a patch where the coordinate functions are cubic polynomialsin u and w. The algebraic form of the bicubic patch is given by

p(u, w) =

3∑

i=0

3∑

j=0

aijuiwj (1.236)

where u, w ∈ [0, 1] and the algebraic coefficients aij are given in vector form as

aij =[

aijxaijy

aijz

](1.237)

For a bicubic patch, there are 16 algebraic coefficients which gives 48 degrees offreedom. The x coordinate of a point on the patch can be written

x = UAxWT (1.238)

50

where U and W are 1 × 4 matrices, and Ax is a 4 × 4 matrix consisting of thex components of the algebraic coefficients

U =[

u3 u2 u 1]

W =[

w3 w2 w 1]

(1.239)

Ax =

a33xa32x

a31xa30x

a23xa22x

a21xa20x

a13xa12x

a11xa10x

a03xa02x

a01xa00x

Similar expressions can be written for the coordinates y and z.

1.3.3 Rational Bezier Surfaces

A rational Bezier surface with control points pi,j , and corresponding weightswi,j , where 0 ≤ i ≤ n and 0 ≤ j ≤ p, is for (u, t) ∈ [0, 1] × [0, 1] defined by

p(u, t) =

∑ni=0

∑pj=0 wi,jpi,jBi,n(u)Bj,p(t)∑n

i=0

∑pj=0 wi,jBi,n(u)Bj,p(t)

(1.240)

where Bi,n(u) and Bj,p(t) are the Bernstein polynomials of degrees n and p, inthe variables u and t, respectively. If wi,j = 0, we replace wi,jpi,j by pi,j . The(n+1) · (p+1) control points form a control point polyhedron. If all weights wi,j

are equal and non-zero, we get an integral Bezier surface as a special case of arational Bezier surface, and Eq.1.240 simplifies to

p(u, t) =n∑

i=0

p∑

j=0

pi,jBi,n(u)Bj,p(t) (1.241)

The coordinate curves of the surface are rational Bezier curves, that is, if we forexample let u = uk be fixed, we get the rational Bezier curve p(t) = p(uk, t),where t ∈ [0, 1]. Figure 1.23 on page 52 shows an example of a rational bicubicBezier surface.

Properties of Rational Bezier Surfaces

Rational Bezier surfaces have some useful properties, similar to the propertiesof rational Bezier curves. The properties are stated in Theorems 1.30–1.32.


A rational Bezier surface p(u, t) with control points p0,0, . . . ,pn,p, and weights

w0,0, . . . , wn,p, where w0,0, wn,0, w0,p, wn,p, are non-zero, satisfies

p(0, 0) = p0,0

p(1, 0) = pn,0

p(0, 1) = p0,p (1.242)

p(1, 1) = pn,p

51

0

1

2

3

4

z

12

34

5

y

12

34

5x

Figure 1.23: Rational bicubic Bezier surface


Every point on a rational Bezier surface lies inside the convex hull of its control

points, provided that all weights wi,j > 0. That is

p(u, t) ∈ CH {p0,0, . . . ,pn,p} , for (u, t) ∈ [0, 1]× [0, 1] (1.243)



reflection, or a combination of these. The transformation of the rational Bezier

surface p(u, t) can be written as a rational Bezier surface where the control

points are transformed by T , that is

T (p(u, t)) =

∑ni=0

∑pj=0 wi,jT (pi,j)Bi,n(u)Bj,p(t)∑n

i=0

∑pj=0 wi,jBi,n(u)Bj,p(t)

(1.244)

This means that instead of transforming each point p(u, t) of a rational Beziersurface, it is sufficient to transform only the control points.

1.3.4 NURBS Surfaces

A NURBS surface with control points pi,j , and corresponding weights wi,j ,where 0 ≤ i ≤ n and 0 ≤ j ≤ p, is for (u, t) ∈ [ud, um−d] × [te, tq−e] defined by

p(u, t) =

∑ni=0

∑pj=0 wi,jpi,jNi,d(u)Nj,e(t)∑n

i=0

∑pj=0 wi,jNi,d(u)Nj,e(t)

(1.245)

where Ni,d(u) and Nj,e(t) are the B-spline basis functions of degrees d and e,with the knot sequences u0, . . . , um and t0, . . . , tq, respectively. If wi,j = 0, wereplace wi,jpi,j by pi,j . The (n + 1) · (p + 1) control points form the controlpoint polyhedron. If all weights wi,j are equal and non-zero, we get an integral

52

–1

–0.5

0

0.5

1

x

–1

–0.5

0

0.5

1

y

–0.5

0

0.5

1

z

Figure 1.24: Unit sphere created as a NURBS surface

B-spline surface as a special case of a NURBS surface, and Eq.1.245 simplifiesto

p(u, t) =

n∑

i=0

p∑

j=0

pi,jNi,d(u)Nj,e(t) (1.246)

The coordinate curves of the surface are NURBS curves, that is, if we for ex-ample let u = uk be fixed, we get the NURBS curve p(t) = p(uk, t), wheret ∈ [te, tq−e].

Figure 1.24 shows an example of a NURBS surface, created as a surface ofrevolution. A unit circle is first created as a NURBS curve with 7 control pointsin the xy-plane (as in Figure 1.20 on page 46). A second unit circle is thencreated as a NURBS curve in the xz-plane. The control points of these twocurves are then combined into a mesh of control points used for the NURBSsurface. The curve in the xz-plane is thus rotated about the z-axis. As can beseen in Figure 1.24, we get the unit sphere centered at the origin.

Figure 1.25 on page 54 shows another example of a NURBS surface, alsocreated as a surface of revolution. A unit circle is first created as a NURBScurve in the xz-plane. The curve is then scaled to a circle of radius 1

4 and thentranslated 1

2 units in the x-direction. The circle is then rotated about the z-axis,and the resulting surface is a torus.

Properties of NURBS Surfaces

NURBS surfaces have some useful properties, stated in Theorems 1.33–1.35.

53

–1

–0.5

0

0.5

1

x

–1

–0.5

0

0.5

1

y

–0.5

0

0.5

1

z

Figure 1.25: Torus created as a NURBS surface


Let a NURBS surface be given by

p(u, t) =

∑ni=0


i=0


(1.247)

Then each segment of the surface is determined by a mesh of (d + 1) · (e + 1)control points. If u ∈ [ur, ur+1) and t ∈ [ts, ts+1), for d ≤ r ≤ m − d − 1 and

e ≤ s ≤ q − e − 1, then

p(u, t) =

∑ri=r−d

∑sj=s−e wi,jpi,jNi,d(u)Nj,e(t)∑r

i=r−d

∑sj=s−e wi,jNi,d(u)Nj,e(t)

(1.248)

That is, to evaluate p(u, t) it is sufficient to evaluate the (d + 1) basis functions

Nr−d,d(u), . . . , Nr,d(u), and the (e+1) basis functions Ns−e,e(t), . . . , Ns,e(t) in-

stead of all (n + 1) · (p + 1) basis functions.


Let a NURBS surface be given by

p(u, t) =

∑ni=0


i=0


(1.249)

If u ∈ [ur, ur+1) and t ∈ [ts, ts+1), for d ≤ r ≤ m − d − 1 and e ≤ s ≤ q − e − 1,and all weights wi,j > 0, then

p(u, t) ∈ CH {pr−d,s−e, . . . ,pr,s} (1.250)

That is, each segment of the surface lies in the convex hull of the control points


54



reflection, or a combination of these. The transformation of the NURBS sur-

face p(u, t) can be written as a NURBS surface where the control points are

transformed by T , that is

T (p(u, t)) =

∑ni=0

∑pj=0 wi,jT (pi,j)Ni,d(u)Nj,e(t)∑n

i=0


(1.251)

This means that instead of transforming each point p(u, t) of a NURBS surface,it is sufficient to transform only the control points.

1.3.5 Approximating Surfaces by Polygon Meshes

In Section 1.2.8 on page 47 we approximated a curve by piecewise linear seg-ments. In a similar way, we can approximate a surface by polygon meshes.Suppose we have defined the surface as a parametric surface, p(u, t), in theparameters u and t. For example, let the surface be given as the Bezier surface

p(u, t) =

n∑

i=0

m∑

j=0

pi,jBi,n(u)Bj,m(t) , (u, t) ∈ [0, 1] × [0, 1] (1.252)

We choose N + 1 values of parameter u, and M + 1 values of parameter t as

ui =i

N, i = 0, 1, . . . , N

tj =j

M, j = 0, 1, . . . , M (1.253)

and evaluate p(u, t) at u = ui and t = tj . We can then form N · M poly-gons, where each polygon Pi,j is defined by the four points p(ui, tj), p(ui+1, tj),p(ui+1, tj+1), and p(ui, tj+1), for i = 0, 1, . . . , N − 1 and j = 0, 1, . . . , M − 1.Figure 1.26 on page 56 shows a surface we want to approximate. We have chosenN = M = 3, and we have evaluated p(u, t) at (N + 1) · (M + 1) = 16 points.This gives us N · M = 9 polygons. The polygon mesh is shown in Figure 1.27on page 56. The polygons in the mesh is in general not planar. A simple wayto get planar polygons is to divide each polygon into triangles. We can do thisby computing a new point qi,j (on the original surface) for polygon Pi,j as

qi,j = p(ui + ui+1

2,tj + tj+1

2) (1.254)

Then we create four triangles with this point and the four points of the polygon,as in Figure 1.28 on page 57, which shows that polygon P1,1 has been triangu-lated. The advantage of using triangles as polygons is that a triangle is planarand convex. Since it is planar, we can compute a normal vector to the planedefined by the triangle. This can be used for hidden surface removal, describedin Section 2.5. It can also be used for adding realism to the image of the surface,by computing a color for each triangle based on the orientation of the triangle

55

Figure 1.26: Surface to be approximated

Figure 1.27: Surface approximated by a polygon mesh

56

Figure 1.28: Polygon mesh with polygon P1,1 triangulated

in relation to various light sources. A convex polygon is desirable when render-ing, because there exist efficient algorithms for rendering convex polygons. Therendering of non-convex polygons is more complicated.

1.4 Representation of 3D Solids

This section briefly describes how solids in 3 dimensions can be represented.Parametric tricubic solids can be used to model the interior of a solid object,and is described in Section 1.4.1. Usually, in computer graphics, we are onlyinterested of the boundary surface of a solid object, since this is what we wouldlike to show as an image.

1.4.1 Parametric Tricubic Solids

The coordinates for a point in a solid can be expressed by three parametricfunctions, one for each of the coordinates x, y, and z. Each function takes threeparameters, u, v, and w.

x = x(u, v, w)

y = y(u, v, w) (1.255)

z = z(u, v, w)

A point in the solid is given by the row vector

p =[

x(u, v, w) y(u, v, w) z(u, v, w)]

(1.256)

57

Algebraic Form

A tricubic solid is given in algebraic form as

p(u, v, w) =

3∑

i=0

3∑

j=0

3∑

k=0

aijkuivjwk , u, v, w ∈ [0, 1] (1.257)

where the algebraic coefficients aijk are given in vector form as

aijk =[

aijkxaijky

aijkz

](1.258)

For a tricubic solid, there are 43 = 64 algebraic coefficients which gives 64 · 3 =192 degrees of freedom. If one of the parameters, for example w, is fixed, we geta surface lying in the solid when we vary u and v. The tricubic solid is boundedby bicubic patches. Tricubic solids are described in more detail in [4].

1.4.2 Solids Bounded by Surfaces

In the previous sections, we have seen surfaces which we can interpret as solids.The ellipsoid in Figure 1.22, the sphere in Figure 1.24, and the torus in Fig-ure 1.25 are some examples.

58

Chapter 2

Visualizing 3D Objects as

2D Projections

2.1 Introduction

This chapter describes the mathematics involved when creating a 2-dimensionalimage from 3-dimensional objects. Transformations in 3D are described in Sec-tion 2.2, followed by projections in Section 2.3. Section 2.4 describes methodsfor specifying what in the 3D space should be seen.

2.2 Transformations

Transformations can be used for changing position, orientation, or shape of anobject. They are also useful for projections, described in Section 2.3. Theterm rigid-body transformation is often used for a transformation which doesnot change the shape of the object.

2.2.1 Translation

A translation is a transformation which moves every point of an object a givendistance in a given direction. The translation can be written by the matrixexpression

p′ = p + t (2.1)

where p′ is the translated point, p is the point to be translated, and t the vectorwith the given distance and direction.

2.2.2 Rotation about a Coordinate Axis

Before deriving the expressions for rotations, we have to define the directionof positive rotation. We define the direction of positive rotation about a coor-dinate axis according to Table 2.1 on page 60. For a right-handed coordinate

59

axis of rotation direction of positive rotationx y → zy z → xz x → y

Table 2.1: Direction of positive rotation

- x

6

y

��

��

��

��

��

��>p

x

y

��

��

��

��

��

��7p′

x′

y′

θ

γ

Figure 2.1: Positive rotation about the z-axis

system, the direction of positive rotation corresponds to a counterclockwise ro-tation when looking at the origin, along the axis of rotation. For a left-handedcoordinate system, positive rotation corresponds to a clockwise rotation. Therotation of a point about the z-axis can be derived as follows. See Figure 2.1.The z-axis is directed out of the page for a right-handed system, and directedinto the page for a left-handed system. In Figure 2.1 on the rotation is positive,and when looking at the origin along the z-axis, the rotation is counterclockwisefor a right-handed system, and clockwise for a left-handed system. We get thefollowing expressions for the coordinates of p and p′

x = |p| cos θ (2.2)

y = |p| sin θ (2.3)

x′ = |p′| cos(θ + γ) (2.4)

y′ = |p′| sin(θ + γ) (2.5)

But |p| = |p′|, and using the addition rules of the sine and cosine functions wecan write Eq. 2.4 and Eq. 2.5 as

x′ = |p|(cos θ cos γ − sin θ sin γ) (2.6)

60

y′ = |p|(sin θ cos γ + cos θ sin γ) (2.7)

Using Eq. 2.2 and Eq. 2.3 in the above equations gives

x′ = x cos γ − y sinγ (2.8)

y′ = y cos γ + x sin γ (2.9)

Since we rotate about the z-axis in a plane parallel to the xy-plane, z′ = z, andwe can write the coordinates of p′ with the matrix expression

[x′ y′ z′

]=[

x y z]

cos γ sin γ 0− sinγ cos γ 0

0 0 1

(2.10)

or more compact asp′ = pRotz,γ (2.11)

In a similar way, we can derive the matrices Rotx,α and Roty,β for rotationsabout the x-axis and the y-axis respectively, as

Rotx,α =

1 0 00 cosα sin α0 − sinα cosα

(2.12)

Roty,β =

cosβ 0 − sinβ0 1 0

sin β 0 cosβ

(2.13)

2.2.3 Scaling about the Origin

An object can be scaled by multiplying the coordinates of every point by thescale factors sx, sy, and sz, all non-zero. The scaling transformation can bewritten

[x′ y′ z′

]=[

x y z]

sx 0 00 sy 00 0 sz

(2.14)

If sx = sy = sz, we get uniform scaling.

2.2.4 Reflections

By using the scaling transformation and choosing combinations of ±1 for sx, sy,and sz, we can reflect a point p in a coordinate plane, in a coordinate axis, or inthe origin. See Figure 2.2 on page 62. p′ is the reflection of p in the xy-plane,p′′ is the reflection of p in the z-axis, and p′′′ is the reflection of p in the origin.The reflected points p′, p′′, and p′′′ are given by the matrix expressions

p′ = pRxy = p

1 0 00 1 00 0 −1

(2.15)

61

HHHHHHHHHHHHHHHHHHj x��* y

6z

HHH��

��HHH

�p

JJ

JJJ

JJ

JJ p′

JJ

JJ

JJ

JJJ]

p′′

�p′′′

Figure 2.2: Reflections in the xy-plane, in the z-axis, and in the origin

p′′ = pRz = p

−1 0 00 −1 00 0 1

(2.16)

p′′′ = pRo = p

−1 0 00 −1 00 0 −1

(2.17)

We can write similar equations as Eq. 2.15 and Eq. 2.16 for reflections in the xz-plane, yz-plane, x-axis, and y-axis respectively. In total we get 8 combinations(including the identity matrix, which gives the point p itself).

2.2.5 Homogeneous Transformations

The transformations rotation, scaling, and reflection can all be expressed usingmatrix multiplication. The composition of any sequence of these transforma-tions can therefore be expressed in a convenient way by a single matrix which isthe product of the matrices for the various transformations. However, transla-tion is expressed by a matrix addition, and a sequence of translation, rotation,scaling, and reflection can not be written by a matrix multiplication. By usinghomogeneous coordinates and homogeneous transformations, we can express allthe transformations above by matrix multiplications. Projective transforma-tions, described in Section 2.3, can also be expressed easily by homogeneoustransformations.

62

Homogeneous Coordinates

In homogeneous coordinates, a point in three dimensions is represented by a vec-tor with four components. A point p with coordinates x, y, and z, representedby the vector

p =[

x y z]

(2.18)

is represented by the homogeneous coordinates hx, hy, hz, and h as

p =[

hx hy hz h]

(2.19)

where h 6= 0. The coordinates x, y, and z are related to the homogeneouscoordinates by

x =hx

h

y =hy

h(2.20)

z =hz

h

The representation in homogeneous coordinates is not unique, since we canchoose any real number h, except 0. For example, the point (1, 2, 3) can berepresented by p1 with h = 1 as

p1 =[

1 2 3 1]

(2.21)

or by p2 using h = 5 as

p2 =[

5 10 15 5]

(2.22)

To convert the homogeneous coordinates back to the ordinary coordinates, wedivide each component by h = 5, and remove the last component to get

p =[

1 2 3]

(2.23)

Usually, we choose h = 1, to simplify computation.Transformations of points in homogeneous coordinates are expressed by 4×4

matrices. For translation, we can write

[x′ y′ z′ 1

]=[

x y z 1]

1 0 0 00 1 0 00 0 1 0tx ty tz 1

(2.24)

Computing this matrix expression gives

[x′ y′ z′ 1

]=[

x + tx y + ty z + tz 1]

(2.25)

Rotation about the x-axis by α can be expressed as

[x′ y′ z′ 1

]=[

x y z 1]Rotx,α (2.26)

63

where Rotx,α is

Rotx,α =

1 0 0 00 cosα sinα 00 − sin α cosα 00 0 0 1

(2.27)

In a similar way, we can write the 4× 4 matrices Roty,β and Rotz,γ for rotatingabout the y-axis and the z-axis respectively

Roty,β =

cosβ 0 − sinβ 00 1 0 0

sin β 0 cosβ 00 0 0 1

(2.28)

Rotz,γ =

cos γ sinγ 0 0− sinγ cos γ 0 0

0 0 1 00 0 0 1

(2.29)

The matrix for scaling, S can be written

S =

sx 0 0 00 sy 0 00 0 sz 00 0 0 1

(2.30)

A composite transformation can be expressed by a single matrix M

M =

a b c pd e f qg i j rk l m h

(2.31)

The upper 3×3 submatrix corresponds to the total effect of rotation and scaling,and the lower 1×3 submatrix corresponds to the total effect of translation. The3 × 1 submatrix can be used for projections, described in Section 2.3.

2.2.6 Rotation about an Arbitrary Axis

Suppose we want to rotate a point p by the angle α about an arbitrary axisdirected from q to r, that is, in the direction r−q, where r 6= q. See Figure 2.3on page 65. The coordinate system is right-handed, and the direction of positiverotation is defined as counterclockwise when looking in the direction from r toq. This definition is consistent with the definition of the direction of positiverotation about a coordinate axis given in Section 2.2.2, if we let q be the origin,and r a point on a positive coordinate axis. Equivalently, the direction ofpositive rotation is the same direction that a right screw has to be turned inorder to be moved from q to r.

64

x

y

z

q

rr − q

bc

p

bc

p′

Figure 2.3: Rotation about an arbitrary axis

We can derive an expression for the rotation by α about r−q as a sequenceof simpler transformations. The idea is to transform the axis of rotation suchthat it is aligned with one of the coordinate axes, then rotate about that axisby α, and then transform the axis back to its original position. We can applythe following 7 transformations:

1. Translate by −q. The translation of the points q and r gives the pointsq′ and r′, such that q′ is at the origin. See Figure 2.4 on page 66.

The point r′ can be expressed by the spherical coordinates

R =√

r′2x + r′2y + r′2z

θ = arctan(r′y , r′x) (2.32)

φ = cos−1 r′zR

where arctan is the two-argument form of tan−1, and is given by

arctan(y, x) =

tan−1 yx , if x > 0

π + tan−1 yx , if x < 0

π2 , if x = 0 and y > 0−π

2 , if x = 0 and y < 0

(2.33)

The angle θ, called azimuth, is the angle between the xz-plane and the planethrough r′ and the z-axis. The angle φ, called colatitude, is the angle betweenr′ and the z-axis.

If r′x = r′y = 0, then r′ lies on the z-axis, and since we assumed that r 6= q,we get r′ 6= q′, and r′ 6= 0, which gives r′z 6= 0 and R = |r′z |, and thus φ = 0 orφ = π. We can then set θ = 0 in step 2, since the axis given by r′ − q′ alreadylies in the xz-plane.

65

x

y

z

q′

r′

θ

φ

Figure 2.4: Axis of rotation translated

2. Rotate by −θ about the z-axis. The rotation of r′ is r′′, which lies in thexz-plane.

3. Rotate by −φ about the y-axis. The rotation of r′′ is r′′′, which lies onthe positive z-axis.

Now the axis of rotation (r′′′ − q′′′) is aligned with the z-axis.

4. Rotate by α about the z-axis.

Now we have to transform the axis of rotation back to the original position.This is done by steps 5–7 below. Note that the transformations is the inversesof the transformations given in step 1–3, in the reverse order.

5. Rotate by φ about the y-axis.

6. Rotate by θ about the z-axis.

7. Translate by q.

By using homogeneous transformations, we can combine the 7 transformationsinto a single transformation, given by the matrix expression

Rot q,r,α = T−qRotz,−θRoty,−φRotz,αRoty,φRotz,θTq (2.34)

By computing this matrix for the given points q and r, and the given angle α,we can save a lot of computations when we want to rotate more than one point

66

u

v

np

p′

projector

projectionplane

Figure 2.5: Planar geometric projection

p. The rotation of p about the axis r − q by α is given by the expression (inhomogeneous coordinates)

p′ = pRot q,r,α (2.35)

2.3 Planar Geometric Projections

This section describes projections from 3 dimensions into 2 dimensions. Theprojections are planar, that is, the projected images reside on a plane rather thana curved surface. The projections are also geometric, which means that eachprojector is a straight line, rather than a general curve. For each point p to beprojected, there is a projector, which is a ray from p intersecting the projectionplane. The intersection point, p′, is the projection of p. See Figure 2.5. Wedefine the projection plane as the uv-plane. The coordinate system given byu,v,n, is a left-handed coordinate system. A left-handed system is often moreintuitive to use, since we can think of u and v as the directions right and up,respectively, and a point p in front of us increases its n coordinate as it movesaway from us.

It is convenient to represent 3D objects in a fixed right-handed coordinatesystem, often called the world coordinate system, and then transform each ob-ject into a view coordinate system given by u,v,n. The position and orientationof the view system relative to the world system can then be varied, in order tocreate different views of the objects in the world. In this section we will as-sume that we have transformed all objects into the view coordinate system.Section 2.4 describes a useful model to transform between the two systems.

67

u

v

n

p

p′

t · DOP

projectionplane

DOP

Figure 2.6: Parallel projection

The projections can be classified into parallel projections and perspectiveprojections, described in Section 2.3.1 and Section 2.3.2, respectively.

2.3.1 Parallel Projections

For parallel projections, all projectors are parallel, and we can denote theirdirection DOP , direction of projection. See Figure 2.6. Parallel projections canbe classified into two types, depending on the relation between DOP and theprojection plane. If DOP is orthogonal to the projection plane, the projection iscalled orthographic, otherwise it is called an oblique projection. We assume thatDOP is not parallel to the projection plane, that is, we assume that DOPn 6= 0.We can write an expression for p′, the projection of p as

p′ = p + t · DOP (2.36)

The point p′ lies in the projection plane, so its n coordinate is 0. We can writeEq. 2.36 as three equations

u′ = u + t · DOPu (2.37)

v′ = v + t · DOPv (2.38)

0 = n + t · DOPn (2.39)

We can solve for t in Eq. 2.39 which gives

t = − n

DOPn(2.40)

68

and then use this t in Eq. 2.37 and Eq. 2.38 which gives

u′ = u − nDOPu

DOPn(2.41)

v′ = v − nDOPv

DOPn(2.42)

We can write this as a matrix expression, and use homogeneous coordinates

p′ = pMpar (2.43)

that is

[u′ v′ 0 1

]=[

u v n 1]

1 0 0 00 1 0 0

−DOPu

DOPn−DOPv

DOPn0 0

0 0 0 1

(2.44)

If DOPu = DOPv = 0, we get orthogonal projection, and Mpar is simplified to

Mpar =

1 0 0 00 1 0 00 0 0 00 0 0 1

(2.45)

2.3.2 Perspective Projections

For perspective projections, all projectors converge into a point called COP ,center of projection. See Figure 2.7 on page 70, which shows the projection ofp into p′ on the projection plane (uv-plane). Let p and COP be given by therow vectors

p =[

u v n]

(2.46)

COP =[

COPu COPv COPn

](2.47)

We assume that COPn < 0 and n > COPn. The projector can be parametrizedas

r(t) = COP(1 − t) + pt (2.48)

The projector intersects the projection plane at t = t′, and the intersectionpoint, p′, is the projection of p, that is

p′ = r(t′) = COP(1 − t′) + pt′ (2.49)

We can write Eq. 2.49 as three equations

u′ = COPu(1 − t′) + ut′ (2.50)

v′ = COPv(1 − t′) + vt′ (2.51)

0 = COPn(1 − t′) + nt′ (2.52)

69

u

v

n

p

p′

projectionplane

COP

Figure 2.7: Perspective projection

Solving for t′ in Eq. 2.52 gives

t′ =COPn

COPn − n, COPn 6= n (2.53)

Using this in Eq. 2.50 and Eq. 2.51, and simplifying, gives

u′ =uCOPn − nCOPu

COPn − n(2.54)

v′ =vCOPn − nCOPv

COPn − n(2.55)

Consider the matrix expression in homogeneous coordinates

[hu′ hv′ 0 h

]=[

u v n 1]

1 0 0 00 1 0 0

−COPu

COPn−COPv

COPn0 − 1

COPn

0 0 0 1

Computing the matrix expression gives

hu′ = u − nCOPu

COPn=

uCOPn − nCOPu

COPn(2.56)

hv′ = v − nCOPv

COPn=

vCOPn − nCOPv

COPn(2.57)

h = 1 − n

COPn=

COPn − n

COPn(2.58)

70

Dividing hu′ and hv′ by h gives

u′ =uCOPn − nCOPu

COPn − n(2.59)

v′ =vCOPn − nCOPv

COPn − n(2.60)

That is, we can express the perspective projection by a matrix multiplicationusing homogeneous coordinates.

Often we let the center of projection be a point on the negative n-axis, thatis, COPu = COPv = 0. Then we can simplify the expressions for u′ and v′ to

u′ = uCOPn

COPn − n(2.61)

v′ = vCOPn

COPn − n(2.62)

The expression COPn

COPn−n is called the foreshortening factor. Since we have chosenCOPn < 0 we can write

COPn

COPn − n=

−|COPn|−|COPn| − n

=|COPn|

|COPn| + n(2.63)

and for n > 0 we get

0 <|COPn|

|COPn| + n< 1 (2.64)

and as an object moves farther away, its projected image becomes smaller. Forn < 0 we can write Eq. 2.63 as

COPn

COPn − n=

|COPn||COPn| − |n| (2.65)

and since we assumed that COPn < n, we get 0 < |COPn| − |n| < |COPn| andthus

|COPn||COPn| − |n| > 1 (2.66)

that is, for an object between the projection plane and COP , the projectedimage becomes magnified.

Pseudodepth

As we have seen before, if we choose COP as a point on the negative n-axis, weget the coordinates of the projected point p′ as

u′ = uCOPn

COPn − n=

u

1 − n/COPn(2.67)

v′ = vCOPn

COPn − n=

v

1 − n/COPn(2.68)

n′ = 0 (2.69)

71

which can be written as a matrix multiplication using homogeneous coordinates

p′ = pMpersp (2.70)

where p′, p, and Mpersp are given by

p′ =[

hu′ hv′ hn′ h]

(2.71)

p =[

u v n 1]

(2.72)

Mpersp =

1 0 0 00 1 0 00 0 0 − 1

COPn

0 0 0 1

(2.73)

The perspective projection in some way destroys information, since we have lostthe distance from the projection plane to the point p. For example, considertwo points, p1 and p2, lying on the same projector, that is, they both projectto the same point p′, and we do not know which of the two points that wasprojected into p′. This can also be seen by the fact that the matrix Mpersp issingular.

It is useful to represent a measure of the distance from the projection planeto the point p. A natural choice of the measure is the expression

n′′ = nCOPn

COPn − n=

n

1 − n/COPn(2.74)

which is called the pseudodepth. Since we have chosen COPn < 0, we can write

n′′ =n

1 + n/|COPn|(2.75)

and as n increases, n′′ increases monotonically. We can then write Eq. 2.70 as

p′ = pMpMproj (2.76)

where the matrix Mp represents a perspective transformation, and is given by

Mp =

1 0 0 00 1 0 00 0 1 − 1

COPn

0 0 0 1

(2.77)

and Mproj is a projection matrix given by

Mproj =

1 0 0 00 1 0 00 0 0 00 0 0 1

(2.78)

The matrix Mproj is the same as the orthogonal projection matrix Mpar givenby Eq. 2.45 on page 69. We can therefore treat a perspective projection as aperspective transformation followed by an orthogonal projection. This fact willbe useful in the clipping process, described in Section 2.4.3. The pseudodepthis useful in for example hidden surface removal, described in Section 2.5.

72

x

y

z

VRPbc

u

UP ,v

VPN ,n

bc wl

bc wr

bc wb

bcwt

e

Figure 2.8: Synthetic-camera

2.4 Viewing in 3D

This section describes a model which is useful for determining what should beseen of the 3D objects. The model is often called the synthetic-camera model.We specify a view volume which determines which objects should be seen. Theprocess of determining which objects that reside in the view volume is calledclipping. The synthetic-camera model, the view volume, and clipping are de-scribed in Section 2.4.1, Section 2.4.2, and Section 2.4.3, respectively.

2.4.1 The Synthetic-Camera Model

We can imagine a camera attached to the view coordinate system, taking snap-shots of objects. The camera can be positioned and oriented by specifyingthe position and orientation of the view coordinate system relative the worldcoordinate system. See Figure 2.8. The position and orientation of the viewcoordinate system are specified by

1. VRP : view reference point, which is the origin of the view coordinatesystem.

2. VPN : viewplane normal, this direction becomes the direction of n in theview coordinate system.

73

3. UP : a vector which defines the up direction, that is, defining v in theview coordinate system. UP is perpendicular to VPN .

VRP , VPN , and UP are specified in the world coordinate system. We cancompute u as the vector product of n and v, that is

u = n× v (2.79)

This gives a left-handed coordinate system with the coordinates u, v, n. In theviewplane, we define a window which determines the size of the snapshots takenby the camera. The window borders are specified in the view coordinate systemby wl, wr, wt, and wb. u = wl and u = wr define the left and right borders ofthe window, and v = wt and v = wb define the top and bottom borders. Wealso define an eye position, e, in the view coordinate system. For a perspectiveprojection, e is the center of projection, COP . For a parallel projection, we candefine the direction of projection, DOP , as the direction of a vector from theorigin of the view coordinate system to the point e.

In an interactive computer graphics application, the following approach isuseful for specifying the position and orientation of the view coordinate systemrelative the world coordinate system.

1. The user specifies the VRP which becomes the origin of the view coordi-nate system.

2. The user specifies VPN , the viewplane normal, or alternatively, a point Swhich the user wants to look at. VPN is then computed as VPN = s− r,where s and r are the position vectors of S and VRP respectively.

The unit vector n is then computed as

n =VPN

|VPN | (2.80)

3. The user specifies an approximate up direction, UP ′, which does not haveto be orthogonal to VPN .

The actual up direction is then computed as

UP = UP ′ − (UP ′ · n)n (2.81)

In this way, UP will be orthogonal to n, provided that UP ′ was not parallel toVPN . Then the unit vector v is formed by

v =UP

|UP | (2.82)

and finally, u is computed asu = n× v (2.83)

This approach gives the user an intuitive way to specify the view.

74

World to View Coordinate System Transformation

The vectors u, v, and n are expressed in the world coordinate system, and aregiven by the row vectors

u =[

ux uy uz

]

v =[

vx vy vz

](2.84)

n =[

nx ny nz

]

We would like to compute a transformation matrix, Awv, which transforms thecoordinates of a point p in the world coordinate system into coordinates in theview coordinate system. Let the view reference point, VRP , be represented bythe row vector r in the world coordinate system. We can write the transforma-tion matrix Awv in homogeneous coordinates as

Awv =

ux vx nx 0uy vy ny 0uz vz nz 0r′x r′y r′z 1

(2.85)

where r′x, r′y, and r′z are given by

r′x = −r · ur′y = −r · v (2.86)

r′z = −r · n

for the details, see [2].

2.4.2 View Volume

The window together with either DOP (for parallel projection), or COP (forperspective projection), define the view volume. For a parallel projection, weget a parallelepiped, see Figure 2.9 on page 76. For simplicity, the figure showsan orthogonal projection, that is, DOP is orthogonal to the viewplane, in thedirection of the negative n-axis. Figure 2.10 shows the view volume for a per-spective projection. For simplicity, we put the COP on the negative n-axis. InFigure 2.9 and Figure 2.10 we have also introduced a front plane at n = F anda back plane at n = B. The front and back planes are used for restricting theview volume, and objects outside the view volume will not be projected. Ingeneral, we assume that en < F < B.

2.4.3 Clipping

Clipping is the process of rejecting objects outside the view volume, and pre-serving objects which reside inside the volume. We can first test which objectsare either completely inside, or completely outside the view volume. An objectpartly inside and partly outside the view volume has to be cut into two pieces,such that one piece is inside, and the other piece is outside the volume.

75

u

v

nbce

DOP

n = F

bc

wl

bc wr

bcwb

bc wt

n = 0 n = B

Figure 2.9: View volume for orthogonal projection

u

v

nbceCOP

n = F

bc

wl

bc wr

bcwb

bc wt

n = 0 n = B

Figure 2.10: View volume for perspective projection

76

v

n

n = 0n = F n = B

bc

COP

v

n

n = 0n = F1−F/COPn

n = B1−B/COPn

Figure 2.11: Prewarping

For an orthogonal projection, the view volume was shown in Figure 2.9 onpage 76. To test if a point p (given in the view coordinate system), is insidethe view volume, we can use the following inequalities, which all must hold

wl < u < wr

wb < v < wt (2.87)

F < n < B

For a perspective projection, the view volume was shown in Figure 2.10 onpage 76. In this case, we can not test each coordinate of p with three indepen-dent inequalities as was the case with the orthogonal projection. However, aswe saw in Section 2.3.2, we can express a perspective projection as a perspectivetransformation followed by an orthogonal projection. The idea is to transformthe point p into a point q (in homogeneous coordinates) as

q = pMp (2.88)

where Mp was given by Eq. 2.77 on page 72. This transformation is often calledprewarping. See Figure 2.11, which shows the original perspective view volumeto the left, and the prewarped view volume to the right. In each part of thefigure, an object is also shown between the viewplane and the back plane. Inthe right part of Figure 2.11, we can see the perspective foreshortening. See [2]for the details of how the volume to the left will be transformed into the volumeto the right. Now we can test if a point p is inside the original view volume bytesting if the prewarped point q is inside the transformed view volume, usingthe following inequalities

wl < u′ < wr

wb < v′ < wt (2.89)

77

F

1 − F/COPn< n′ <

B

1 − B/COPn

where n′ is the pseudodepth of the point p.Two common clipping algorithms are the Cohen-Sutherland line clipping al-

gorithm, and the Liang-Barsky polygon clipping algorithm. They are describedin [1].

2.5 Hidden Surface Removal

In order to get a realistic view of a 3D object, we need to remove hidden surfaces.In this section we mention some common methods. For more details, see [1] and[2].

2.5.1 Back Face Culling

For a planar polygon, we can define a unique normal vector. It is common tospecify the vertices of a polygon in a counterclockwise order, and define the nor-mal vector as pointing towards us. When we have set up the viewing transformand transformed each vertex of each polygon, we can check the direction of thenormal vectors. If a polygon has a normal vector pointing away from us, thepolygon can be culled out (removed) since it can not be seen. However, thismethod is not guaranteed to work correctly for a nonconvex object.

2.5.2 Depth Buffer Algorithm

The depth buffer algorithm is used after projection, during the rendering ofa polygon. The pseudodepth is stored for each vertex in the polygon. Theidea is to render the polygon without showing it on the screen directly, butto store the color of each point of the polygon into an intermediate buffer andto store the pseudodepth of the point in a depth buffer if the current value inthe depth buffer corresponds to a point of a polygon which are more distant.If the current value in the depth buffer corresponds to a point of a previouslyrendered polygon with a smaller pseudodepth, the intermediate buffer and thedepth buffer are not modified. When all polygons have been rendered into theintermediate buffer, the contents of the buffer is shown on the screen.

78

Chapter 3

Rendering the Projected

Objects

3.1 Introduction

This chapter describes how the images of the projected objects are rendered,that is, plotted on a computer screen. Nowadays, raster displays are mostcommon. A raster display consists of a rectangular grid of pixels. Pixel is anabbreviation for picture element. Rendering of curves is described in Section 3.2and rendering of surfaces in Section 3.3.

3.2 Rendering Curves

In this section, we will look at the basic algorithms for rendering curves effi-ciently. Section 3.2.1 and Section 3.2.2 describes rendering of lines and circlesrespectively. Rendering of Bezier curves is described briefly in Section 3.2.3.

3.2.1 Rendering Lines

In this section we derive an algorithm for rendering lines, the midpoint line

algorithm. The algorithm generates the pixels which approximate a line betweentwo end points, given by integer coordinates.

The Midpoint Line Algorithm

Before we derive the algorithm, consider the line between the points p = (0, 0)and q = (7, 3). Figure 3.1 on page 80 shows the ideal line, and the pixels whichapproximate the line. Note, in this case, we have plotted exactly one pixel ineach column of the grid. For this particular line, the algorithm starts by plottingthe pixel at (0, 0), increases the x coordinate in steps of 1, and plots the pixelclosest to the ideal line. This works because the slope of the line, which is 3

7 , is

79

0

1

2

3

0 1 2 3 4 5 6 7

Figure 3.1: Midpoint line algorithm

less than 1. If the slope was greater than 1, we would get more than one pixelin some of the columns.

To derive the midpoint line algorithm for a line between the points (x0, y0)and (xn, yn), we start with the following assumptions

x0 < xn (3.1)

0 ≤ yn − y0

xn − x0=

∆y

∆x≤ 1 (3.2)

The algorithm can be extended to handle general lines, this which will be dis-cussed briefly later.

Consider Figure 3.2 on page 81, which shows an intermediate step in thealgorithm. We have plotted a pixel at (xp, yp), denoted the previous pixel, andwe want to choose the current pixel to be plotted. For the current pixel we gettwo choices, denoted by E (east), and NE (northeast). Let Q be the point ofintersection between the ideal line and the grid line x = xp + 1. Let M be themidpoint between the pixels E and NE , that is, M = (xp + 1, yp + 1/2). Tochoose between E and NE , we check which side of the line M lies. If M is belowthe line, as is the case in Figure 3.2, we choose the current pixel as NE . If M isabove the line, we choose E. If M is exactly on the line, we can choose either Eor NE , so we choose E arbitrarily. We can represent the line with an implicitfunction as

F (x, y) = ax + by + c = 0 (3.3)

and also in explicit form as

y =∆y

∆xx + B (3.4)

which can be written∆y

∆xx − y + B = 0 (3.5)

multiplying by ∆x gives

∆y · x − ∆x · y + ∆x · B = 0 (3.6)

80

yp

yp + 1

xp xp + 1

previous

E

NE

current

M

Q

ME

MNE

next

Figure 3.2: Choices for current and next pixels

comparing with Eq. 3.3 gives

a = ∆y

b = −∆x (3.7)

c = ∆x · B

We note that F (x, y) < 0 for a point above the line, F (x, y) > 0 for a pointbelow the line, and F (x, y) = 0 for a point on the line. We check if the midpointM = (xp + 1, yp + 1/2) is above or below the line by introducing a decisionvariable d, defined by

d = F (xp + 1, yp + 1/2) = a(xp + 1) + b(yp + 1/2) + c (3.8)

If d > 0, M is below the line, and we choose the current pixel as NE . If d < 0,M is above the line, and we choose the pixel E. If d = 0, M is on the line, sowe choose E.

When we have chosen the current pixel, we update the decision variable.The new value depends on which pixel we have chosen, and we get two cases:

1. If we have chosen NE , we move the midpoint M one step in both thex and the y direction, that is, to MNE . Thus, we get the new decisionvariable, dnew , as

dnew = F (xp + 2, yp + 3/2) = a(xp + 2) + b(yp + 3/2) + c (3.9)

which can be written

dnew = a(xp + 1) + b(yp + 1/2) + c + a + b (3.10)

81

comparing with Eq. 3.8, denoting the old value of d as dold , and usingEq. 3.7, gives

dnew = dold + a + b = dold + ∆y − ∆x (3.11)

We denote the difference between dnew and dold when NE is chosen as∆NE , given by

∆NE = ∆y − ∆x (3.12)

2. If E was chosen, the midpoint M is moved one step in the x direction,that is, to ME , and we get dnew as

dnew = F (xp + 2, yp + 1/2) = a(xp + 2) + b(yp + 1/2) + c (3.13)

which is written

dnew = a(xp + 1) + b(yp + 1/2) + c + a (3.14)

comparing with Eq. 3.8, and using Eq. 3.7, gives

dnew = dold + a = dold + ∆y (3.15)

and the difference between dnew and dold when E was chosen is denoted∆E , given by

∆E = ∆y (3.16)

Finally, we need an expression for the initial value of d, denoted dinit , computedat the midpoint M when the first point at (x0, y0) has been plotted

dinit = F (x0 + 1, y0 + 1/2) = a(x0 + 1) + b(y0 + 1/2) + c (3.17)

which is written

dinit = ax0 + by0 + c + a +b

2= F (x0, y0) + a +

b

2(3.18)

but F (x0, y0) = 0, since the point (x0, y0) lies on the line, and we get

dinit = a +b

2= ∆y − ∆x

2(3.19)

We can refine the algorithm to work with integers only by defining a new decisionvariable d′, as

d′ = 2d = 2F (xp + 1, yp + 1/2) (3.20)

Since we only use the checks d < 0, d > 0, and d = 0, the same checks willapply to the new decision variable d′. The initial value of d′ is then

d′init = 2∆y − ∆x (3.21)

and the increments of d′ when NE and E are chosen, respectively, are given by

∆′

NE = 2(∆y − ∆x)

∆′

E = 2∆y (3.22)

82

The advantage is that the algorithm now only uses integer addition and subtrac-tion, and multiplications by 2, which can be implemented efficiently in hardware.

We have only treated the special case when the slope of the line is between0 and 1, and we assumed that x0 < xn. We can extend the algorithm to handlemore general cases, for example, if the slope is > 1, we can exchange the roleof x and y in the algorithm, such that we increase y in steps of 1, and choosebetween pixels with different x coordinates, that is, we choose between the pixelsN (north) and NE (northeast).

For cases where x0 > xn, we can decrease x, and adjust the the algorithmto choose between pixels W (west) and NW (northwest) or SW (southwest)respectively. See Section 4.2 for a program that handles general lines.

3.2.2 Rendering Circles

In this section we derive an algorithm for rendering circles, the midpoint circle

algorithm. A similar technique for choosing between pixels is used as was usedin the midpoint line algorithm.

The Midpoint Circle Algorithm

We assume that the circle has an integer radius, R, and is centered at the origin.For a circle not centered at the origin, we can translate each pixel the algorithmgenerates by (xc, yc), where (xc, yc) is the integer coordinates of the center ofthe circle.

The algorithm basically generates pixels in the second octant (here, by octantwe mean a quadrant divided into two parts), and uses symmetry to generatethe pixels in the remaining 7 octants. Before we derive the algorithm, considerthe circle with radius 15 centered at the origin. Figure 3.3 on page 84 showsthe part of the ideal circle and the pixels generated by the algorithm in the firstquadrant. The algorithm starts by plotting the pixel at (x, y) = (0, R), thenchooses between the pixels (1, R) and (1, R−1), and plots the pixel closest to theideal circle. It repeats this process as long as y > x. For each pixel generated,8 pixels are plotted, one for each octant.

Consider Figure 3.4 on page 84, which shows a magnified view of an in-termediate step in the algorithm. We have plotted a pixel at (xp, yp), de-noted the previous pixel, and we want to choose the current pixel to be plot-ted. For the current pixel we get two choices, denoted by E (east), and SE

(southeast). To choose between these pixels, we check whether the midpointM = (xp +1, yp −1/2) is inside or outside the ideal circle. If M is outside, as inFigure 3.4, the pixel SE is chosen, since it is closest to the circle. If M is inside,pixel E is chosen, and if M is exactly on the circle, we can choose either E orSE , so we choose for example SE . To check whether M is inside or outside thecircle, we can use the implicit equation of a circle, given by

F (x, y) = x2 + y2 − R2 = 0 (3.23)

For a point inside the circle, F (x, y) < 0, and for a point outside the circle,F (x, y) > 0. If the point lies on the circle, F (x, y) = 0. We define a decision

83

x0 15

y

0

15

Second octant

First octant

Figure 3.3: Midpoint circle algorithm

yp − 1

yp

xp xp + 1

previous

E

SE

current

M ME

MSE

next

Figure 3.4: Choices for current and next pixels

84

variable, d, corresponding to the midpoint M , as

d = F (xp + 1, yp − 1/2) = (xp + 1)2 + (yp − 1/2)2 − R2 (3.24)

If d < 0, M is inside the circle, so we choose pixel E. If d ≥ 0, M is outside oron the circle, so we choose pixel SE . When we have chosen the current pixel,we update the decision variable to correspond to one of the midpoints ME andMSE respectively. In Figure 3.4 we choose the current pixel as SE , so the nextmidpoint will be MSE . We denote the old decision variable given by Eq. 3.24as dold . Depending on the pixel chosen as the current we get two cases:

1. If E is chosen, we get the new decision variable dnew , as

dnew = F (xp + 2, yp − 1/2) = (xp + 2)2 + (yp − 1/2)2 − R2 (3.25)


dnew = (xp + 1)2 + (yp − 1/2)2 − R2 + 2xp + 3 (3.26)

which is then written

dnew = dold + 2xp + 3 (3.27)

The increment of d when E is chosen is denoted ∆E , and is given by

∆E = 2xp + 3 (3.28)

2. If SE is chosen, we get the new decision variable dnew , as

dnew = F (xp + 2, yp − 3/2) = (xp + 2)2 + (yp − 3/2)2 − R2 (3.29)


dnew = (xp + 1)2 + (yp − 1/2)2 − R2 + 2xp + 3 − 2yp +8

4(3.30)

and thusdnew = dold + 2xp − 2yp + 5 (3.31)

The increment of d when SE is chosen is denoted ∆SE , and is given by

∆SE = 2xp − 2yp + 5 (3.32)

Finally, we need an initial value of d, denoted dinit , corresponding to the firstmidpoint M = (1, R − 1/2)

dinit = F (1, R − 1/2) = 12 + (R − 1/2)2 − R2 =5

4− R (3.33)

We can refine the algorithm to use integers only by defining a new decisionvariable, d′, given by

d′ = d − 1

4(3.34)

85

The initial value of d′, denoted d′init , is then

d′init = 1 − R (3.35)

The original comparisons d < 0 and d ≥ 0 then becomes d′ < − 14 and d ≥ − 1

4 ,but since d′ is initialized to d′init , which is an integer, and the increments ∆E

and ∆SE are both integers, d′ will always be an integer, so we can use thecomparisons d′ < 0 and d′ ≥ 0 instead.

See Section 4.2 for a program that implements the midpoint circle algorithm.

3.2.3 Rendering Bezier Curves

In this section we describe how Bezier curves can be rendered. One way todo this is to approximate the Bezier curve by piecewise linear segments, as inSection 1.2.8 on page 47. However, it can be difficult to choose the spacingbetween the parameter values ui in order to get a good approximation, and atthe same time get a fast rendering.

In Section 1.2.3 on page 29 subdivision of a Bezier curve was described. Weused the de Casteljau algorithm to compute the control points for two Beziercurves created by subdividing a given Bezier curve. For example, Figure 1.15on page 36 showed a cubic Bezier curve divided into two Bezier curves. Notethat the convex hull for each of the two parts of the original curve was shrinkedcompared to the convex hull for the original curve. We can use the techniqueof subdivision to render a Bezier curve by recursively subdivide the curve, untilthe convex hull for each part of the curve is sufficiently small. See Section 4.3for a program implementing this idea.

3.3 Rendering Surfaces

A parametric surface can be rendered by rendering the coordinate curves of thesurface. For example, in Figure 1.23 on page 52, a rational Bezier surface wasshown. By choosing the number of coordinate curves sufficiently large, we canget the impression of a surface. Figure 1.24 and Figure 1.25 on pages 53 and 54,respectively, are other examples.

In Section 1.3.5 on page 55 approximation of surfaces by polygon mesheswas described. The polygons in the mesh were triangulated. In this way, thepolygons are both convex and planar, and there exist efficient algorithms forrendering them. Figure 3.5 on page 87 shows an approximation of a spherecreated by a polyhedron consisting of 320 polygons. The polyhedron was createdby triangulization of an icosahedron. An icosahedron is a regular polyhedronwith 20 triangular faces. Each face was divided into 4 triangles, yielding apolyhedron with 80 triangular faces. Each of these faces was then further dividedinto 4 triangles, thus giving 320 triangles. Finally, each vertex of the polyhedronwas translated to get a unit distance to the origin.

86

–1

–0.5

0

0.5

1

x

–1

–0.5

0

0.5

1

y

–1

–0.5

0

0.5

1

z

Figure 3.5: Approximation of a sphere by polygons

87

Chapter 4

Programming Examples

4.1 Introduction

This chapter shows some program examples written in Java. Section 4.2 demon-strates the midpoint algorithms for lines, circles and for ellipses. In order to seethe pixels chosen by the algorithms, we simulate the pixels as rather large cir-cles. Section 4.3 lists a program for rendering a Bezier curve, using recursivesubdivision. In this program, we use the line drawing routines supported inJava. Note, these programs are meant to demonstrate the basic ideas only. Forreal graphics applications, we should use graphics libraries such as OpenGL(http://www.opengl.org), for example.

4.2 Midpoint Algorithm for Lines, Circles and

Ellipses

4.2.1 Midpoint Line Algorithm

//

// Demonstration of the Midpoint Line Algorithm

//

import java.awt.*;

import java.awt.event.*;

public class MidpointLine extends Frame {

public static void main(String[] args) {

new MidpointLine();

}

public MidpointLine() {

88

super("Demo of the Midpoint Line Algorithm");

addWindowListener(new WindowAdapter()

{public void windowClosing(WindowEvent e){System.exit(0);}});

setSize(500,510);

add("Center", new MyCanvas());

setCursor(Cursor.getPredefinedCursor(Cursor.CROSSHAIR_CURSOR));

show();

}

public class MyCanvas extends Canvas implements MouseListener {

final int GRID_SPACE = 15;

final int PIXEL_RADIUS = 6;

int maxX, maxY;

int gridX0, gridY0;

int P1x, P1y;

int P2x, P2y;

int nPoints = 0;

public MyCanvas() {

addMouseListener(this);

}

public void initCanvas() {

Dimension d = getSize();

maxX = d.width - 1;

maxY = d.height - 1;

gridX0 = GRID_SPACE;

gridY0 = GRID_SPACE*(maxY/GRID_SPACE);

}

public void drawGrid(Graphics g) {

int x, y;

g.setColor(Color.black);

for (x = GRID_SPACE; x < maxX; x += GRID_SPACE) {

g.drawLine(x, GRID_SPACE,

x, GRID_SPACE*(maxY/GRID_SPACE));

}

for (y = GRID_SPACE; y < maxY; y += GRID_SPACE) {

g.drawLine(GRID_SPACE, y,

GRID_SPACE*(maxX/GRID_SPACE), y);

}

}

public void drawPixel(Graphics g, int x, int y) {

// Draw the simulated pixel as a filled circle on the grid

// Convert from grid coordinates to Canvas coordinates

89

int xc = gridX0 + x * GRID_SPACE;

int yc = gridY0 - y * GRID_SPACE;


g.fillOval(xc - PIXEL_RADIUS, yc - PIXEL_RADIUS,

2 * PIXEL_RADIUS, 2 * PIXEL_RADIUS);

}

public void drawMidpointLine(Graphics g, int x1, int y1,

int x2, int y2) {

int dx = Math.abs(x2 - x1);

int dy = Math.abs(y2 - y1);

int d;

int deltaE = 2 * dy;

int deltaN = 2 * dx;

int deltaNE;

if (dx > dy) {

d = 2 * dy - dx;

deltaNE = 2 * (dy - dx);

} else {

d = 2 * dx - dy;

deltaNE = 2 * (dx - dy);

}

if (x1 > x2) {

int t = x1; x1 = x2; x2 = t;

t = y1; y1 = y2; y2 = t;

}

int x = x1;

int y = y1;

int yInc = y2 < y1 ? -1 : 1;

drawPixel(g, x, y);

if (dx > dy) {

while (x != x2) {

if (d <= 0) {

d += deltaE;

} else {

d += deltaNE;

y+=yInc;

}

x++;

drawPixel(g, x, y);

}

} else {

while (y != y2) {

if (d <= 0) {

d += deltaN;

} else {

d += deltaNE;

x++;

}

y+=yInc;

drawPixel(g, x, y);

90

}

}

}

public void paint(Graphics g) {

initCanvas();

drawGrid(g);

if (nPoints == 1) {

drawPixel(g, P1x, P1y);

} else if (nPoints == 2) {

// Draw ideal line

int xIdeal1 = gridX0 + P1x * GRID_SPACE;

int yIdeal1 = gridY0 - P1y * GRID_SPACE;

int xIdeal2 = gridX0 + P2x * GRID_SPACE;

int yIdeal2 = gridY0 - P2y * GRID_SPACE;

g.setColor(Color.red);

g.drawLine(xIdeal1, yIdeal1, xIdeal2, yIdeal2);

drawMidpointLine(g, P1x, P1y, P2x, P2y);

}

}

public void mouseClicked(MouseEvent e) {}

public void mouseEntered(MouseEvent e) {}

public void mouseExited(MouseEvent e) {}

public void mouseReleased(MouseEvent e) {}

public void mousePressed(MouseEvent e) {

if (nPoints == 2) {

nPoints = 0;

}

if (nPoints == 0) {

// Get first end point of the line

// Convert to grid coordinates

P1x = Math.round((e.getX()-gridX0)/((float)GRID_SPACE));

P1y = -Math.round((e.getY()-gridY0)/((float)GRID_SPACE));

nPoints++;

repaint();


// Get second end point of the line




nPoints++;

repaint();

}

}

}

}

91

4.2.2 Midpoint Circle Algorithm

//

// Demonstration of the Midpoint Circle Algorithm

//

import java.awt.*;


public class MidpointCircle extends Frame {


new MidpointCircle();

}

public MidpointCircle() {

super("Demo of the Midpoint Circle Algorithm");



setSize(500,510);



show();

}




int maxX, maxY;

int gridX0, gridY0;

int P1x, P1y;

int P2x, P2y;

int R;

int nPoints = 0;

public MyCanvas() {


}



maxX = d.width - 1;




}

92


int x, y;





}




}

}









}

public void drawCirclePixels(Graphics g, int x, int y,

int xc, int yc) {

drawPixel(g, xc + y, yc + x); // Octant 1

drawPixel(g, xc + x, yc + y); // Octant 2

drawPixel(g, xc - x, yc + y); // Octant 3

drawPixel(g, xc - y, yc + x); // Octant 4

drawPixel(g, xc - y, yc - x); // Octant 5

drawPixel(g, xc - x, yc - y); // Octant 6

drawPixel(g, xc + x, yc - y); // Octant 7

drawPixel(g, xc + y, yc - x); // Octant 8

}

public void drawMidpointCircle(Graphics g,

int xc, int yc, int R) {

int x = 0;

int y = R;

int d = 1 - R;

drawCirclePixels(g, x, y, xc, yc);

while (y > x) {

if (d < 0) {

d += 2 * x + 3;

x++;

} else {

93

d += 2 * (x - y) + 5;

x++;

y--;

}

drawCirclePixels(g, x, y, xc, yc);

}

}


initCanvas();

drawGrid(g);

if (nPoints == 1) {



// Draw ideal circle

int xIdeal = gridX0 + P1x * GRID_SPACE;

int yIdeal = gridY0 - P1y * GRID_SPACE;

int rIdeal = R * GRID_SPACE;


g.drawOval(xIdeal - rIdeal, yIdeal - rIdeal,

2 * rIdeal, 2 * rIdeal);


drawMidpointCircle(g, P1x, P1y, R);

}

}






if (nPoints == 2) {

nPoints = 0;

}

if (nPoints == 0) {

// Get center of circle




nPoints++;

repaint();


// Get a point, compute integer radius




float dx = P2x - P1x;

float dy = P2y - P1y;

R = Math.round((float)Math.sqrt(dx * dx + dy * dy));

94

nPoints++;

repaint();

}

}

}

}

4.2.3 Midpoint Ellipse Algorithm

//

// Demonstration of the Midpoint Ellipse Algorithm

//

import java.awt.*;


public class MidpointEllipse extends Frame {


new MidpointEllipse();

}

public MidpointEllipse() {

super("Demo of the Midpoint Ellipse Algorithm");



setSize(500,510);



show();

}




int maxX, maxY;

int gridX0, gridY0;

int P1x, P1y;

int P2x, P2y;

int a, b;

int nPoints = 0;

public MyCanvas() {


}


95


maxX = d.width - 1;




}


int x, y;





}




}

}









}

public void drawEllipsePixels(Graphics g, int x, int y,

int xc, int yc) {

drawPixel(g, xc + x, yc + y); // Quadrant 1

drawPixel(g, xc - x, yc + y); // Quadrant 2

drawPixel(g, xc - x, yc - y); // Quadrant 3

drawPixel(g, xc + x, yc - y); // Quadrant 4

}

public void drawMidpointEllipse(Graphics g, int xc, int yc,

int a, int b) {

int x = 0;

int y = b;

int a2 = a * a;

int b2 = b * b;

double d1 = b2 - a2*b + 0.25*a2;

drawEllipsePixels(g, x, y, xc, yc);

96

while (a2*(y-0.5) > b2*(x+1)) {

if (d1 < 0) {

d1 += b2*(2*x+3);

x++;

} else {

d1 += b2*(2*x+3)+a2*(2-2*y);

x++;

y--;

}


}

double d2 = b2*(x+0.5)*(x+0.5) + a2*(y-1)*(y-1) - a2*b2;

while(y > 0) {

if (d2 < 0) {

d2 += b2*(2*x+2)+a2*(3-2*y);

x++;

y--;

} else {

d2 += a2*(3-2*y);

y--;

}


}

}


initCanvas();

drawGrid(g);

if (nPoints == 1) {



// Draw ideal ellipse

int xIdeal = gridX0 + P1x * GRID_SPACE;

int yIdeal = gridY0 - P1y * GRID_SPACE;

int aIdeal = Math.abs(P1x - P2x) * GRID_SPACE;

int bIdeal = Math.abs(P1y - P2y) * GRID_SPACE;


g.drawOval(xIdeal-aIdeal, yIdeal-bIdeal,

2*aIdeal, 2*bIdeal);

int a = Math.abs(P1x - P2x);

int b = Math.abs(P1y - P2y);

drawMidpointEllipse(g, P1x, P1y, a, b);

}

}





97


if (nPoints == 2) {

nPoints = 0;

}

if (nPoints == 0) {

// Get coordinates of the center




nPoints++;

repaint();


// Get coordinates of a point

// of the bounding box




nPoints++;

repaint();

}

}

}

}

4.3 Algorithm for Bezier Curve

//

// Representation of a 2D Point

//

public class Point2D {

float x;

float y;

public Point2D(float x, float y) {

this.x = x;

this.y = y;

}

}

98

//

// Representation of a Bezier Curve

//

import java.awt.*;

public class Bezier {

private Point2D[] ctrlPoints;

private float xExtent, yExtent, pixelSize;

private int centerX, centerY, nPoints;

private Graphics g;

private Bezier left, right;

public Bezier(Graphics g, Point2D[] ctrlPoints,

float pixelSize, int centerX, int centerY) {

this.ctrlPoints = ctrlPoints;

this.nPoints = ctrlPoints.length;

this.g = g;

this.pixelSize = pixelSize;

this.centerX = centerX;

this.centerY = centerY;

float xMin, xMax, yMin, yMax;

Point2D P = ctrlPoints[0];

xMin = xMax = P.x;

yMin = yMax = P.y;

for (int i = 1; i < nPoints; i++) {

P = ctrlPoints[i];

xMin = P.x < xMin ? P.x : xMin;

yMin = P.y < yMin ? P.y : yMin;

xMax = P.x > xMax ? P.x : xMax;

yMax = P.y > yMax ? P.y : yMax;

}

this.xExtent = Math.abs(xMax - xMin);

this.yExtent = Math.abs(yMax - yMin);

}

public void draw(float limit) {

Point2D P1 = ctrlPoints[0];

Point2D P2 = ctrlPoints[nPoints-1];

if (nPoints == 2 || xExtent < limit && yExtent < limit) {

g.drawLine(intX(P1.x),

intY(P1.y),

intX(P2.x),

intY(P2.y));

} else {

subdivide();

left.draw(limit);

right.draw(limit);

99

}

}

private int intX(float x) {

return Math.round(centerX + x/pixelSize);

}

private int intY(float y) {

return Math.round(centerY - y/pixelSize);

}

private void subdivide() {

int i, j;

Point2D[] leftPoints = new Point2D[nPoints];

Point2D[] rightPoints = new Point2D[nPoints];

Point2D[] tempPoints = new Point2D[nPoints];

Point2D P, P1, P2;

for (i = 0; i < nPoints; i++) {

P = ctrlPoints[i];

tempPoints[i] = new Point2D(P.x, P.y);

}

P = tempPoints[0];

leftPoints[0] = new Point2D(P.x, P.y);

P = tempPoints[nPoints-1];

rightPoints[nPoints-1] = new Point2D(P.x, P.y);

for (j = 1; j < nPoints; j++) {

for (i = 0; i < nPoints - j; i++) {

tempPoints[i].x += tempPoints[i+1].x;

tempPoints[i].x *= 0.5;

tempPoints[i].y += tempPoints[i+1].y;

tempPoints[i].y *= 0.5;

}

P1 = tempPoints[0];

leftPoints[j] = new Point2D(P1.x, P1.y);

P2 = tempPoints[nPoints-1-j];

rightPoints[nPoints-1-j] = new Point2D(P2.x, P2.y);

}

left = new Bezier(g, leftPoints, pixelSize, centerX, centerY);

right = new Bezier(g, rightPoints, pixelSize, centerX, centerY);

}

}

100

//

// Demonstration of Rendering Bezier Curves

//

import java.awt.*;


public class BezierDemo extends Frame {

private Bezier curve;


new BezierDemo();

}

public BezierDemo() {

super("Demo of Bezier Curves");



setSize(500,500);



show();

}


final static float REAL_X = 10.0f;

final static float REAL_Y = 10.0f;

final static int MAX_POINTS = 20;

float pixelSize;

int maxX, maxY;

int centerX, centerY;

Point2D[] ctrlPoints = new Point2D[MAX_POINTS];

int nPoints = 0;

boolean lastPoint = false;

public MyCanvas() {


}



maxX = d.width - 1;


centerX = maxX / 2;

centerY = maxY / 2;

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pixelSize = Math.max(REAL_X/maxX, REAL_Y/maxY);

}


int i;

Point2D P;

initCanvas();


for (i = 0; i < nPoints; i++) {

P = ctrlPoints[i];

g.drawRect(intX(P.x)-2,

intY(P.y)-2,4,4);

if (i > 0) {

g.drawLine(intX(ctrlPoints[i-1].x),

intY(ctrlPoints[i-1].y),

intX(ctrlPoints[i].x),

intY(ctrlPoints[i].y));

}

}

if (nPoints > 1 && lastPoint) {


Point2D[] points = new Point2D[nPoints];

for (i = 0; i < nPoints; i++)

points[i] = ctrlPoints[i];

curve = new Bezier(g, points, pixelSize,

centerX, centerY);

curve.draw(pixelSize*2);

}

}






if (lastPoint) {

lastPoint = false;

nPoints = 0;

}

if (nPoints<MAX_POINTS) {

Point2D P = new Point2D(floatX(e.getX()),

floatY(e.getY()));

ctrlPoints[nPoints] = P;

nPoints++;

}

if (e.getButton() == MouseEvent.BUTTON2) {

lastPoint = true;

}

repaint();

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}

public int intX(float x) {

return Math.round(centerX + x/pixelSize);

}

public int intY(float y) {

return Math.round(centerY - y/pixelSize);

}

public float floatX(int x) {

return (x - centerX) * pixelSize;

}

public float floatY(int y) {

return -(y - centerY) * pixelSize;

}

}

}

103

Conclusion

Computer graphics is a broad subject, and during the work with this paperI have only scratched the surface. I have focused on the mathematics usedfor representing and transforming three-dimensional objects. There is a lot towrite about computer graphics algorithms, which are also very interesting. Thedevelopments in this area are rapid, especially of computer graphics hardware.Computer graphics are used in many different areas, such as CAD systems,computer games, movies, and flight simulators.

During the work I have learned a lot, but I have also realized that there is alot more to learn about computer graphics. There are many good books aboutthe subject. The books listed in the bibliography on page 105 serve as a goodstarting point.

104

Bibliography

[1] Foley, James D et al, Computer Graphics: Principles and Practice. 2nd ed.in C. Addison-Wesley. 2000.

[2] Hill, Francis S., Computer Graphics. Macmillan. 1990.

[3] Marsh, Duncan, Applied Geometry for Computer Graphics and CAD.Springer. 1999.

[4] Mortenson, Michael E., Geometric modeling. 2nd ed. John Wiley & sons.1997.

[5] Roe, John, Elementary Geometry. Oxford. 1995.

[1] is a very good book, containing various aspects of computer graphics. Itis useful both as an introduction, but also as an advanced text. It covers rep-resentation of graphical objects, rendering algorithms, design of user interfaces,and the history of computer graphics. [2] is also very good. It is not as completeas [1], but perphaps easier to understand. [3] is a good introduction to Beziercurves and B-splines. [4] describes parametric curves, surfaces, and solids ingeneral.

105

Biography

My name is Lennart Jordansson, and I grew up in Saffle in Varmland. I receivedthe degree of M. Sc. (civilingenjor) in computer science and electrical engineer-ing at Lulea University of Technology in 1999. I have worked as a hardware andsoftware engineer in Karlstad for about four years. My interests include elec-tronics, computer science, mathematics, and computer graphics. I went back toLulea to study mathematics in 2002, and this work is a result of combining someof my interests. It has been a great opportunity to learn more about computergraphics and mathematics.

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