Computing & Information Sciences Kansas State University Lecture 27 of 42CIS 636/736: (Introduction...

Computing & Information SciencesKansas State University

Lecture 27 of 42CIS 636/736: (Introduction to) Computer Graphics

Lecture 29 of 42

Wednesday, 02 April 2008

William H. Hsu

Department of Computing and Information Sciences, KSU

KSOL course pages: http://snipurl.com/1y5gc

Course web site: http://www.kddresearch.org/Courses/CIS636

Instructor home page: http://www.cis.ksu.edu/~bhsu

Readings:

Sections 11.1 – 11.6, Eberly 2e – see http://snurl.com/1ye72

http://graphics.ucsd.edu/courses/cse167_f06/

Bezier Curves and Splines



Cubic CurvesCubic Curves

CSE167: Computer Graphics

Instructor: Steve Rotenberg

UCSD, Fall 2006

© 2006 Rotenberg, S., University of California San Diego




Understanding and UsingOpenGL EvalMesh/EvalCurve

Understanding and UsingOpenGL EvalMesh/EvalCurve

Render a smooth, cubic surface by tessellating it into a set of triangles See: evalMesh*d

© 2006 Rotenberg, S., University of California San Diego




Polynomial FunctionsPolynomial Functions

Linear:

Quadratic:

Cubic:

dctbtattf

cbtattf

battf

23

2



Vector Polynomials (Curves)Vector Polynomials (Curves)

Linear:

Quadratic:

Cubic:

We usually define the curve for 0 ≤ t ≤ 1

dcbaf

cbaf

baf

tttt

ttt

tt

23

2



Linear InterpolationLinear Interpolation

Linear interpolation (Lerp) is a common technique for generating a new value that is somewhere in between two other values

A ‘value’ could be a number, vector, color, or even something more complex like an entire 3D object…

Consider interpolating between two points a and b by some parameter t

baba tttLerp 1,,

a

b

t=1.

. 0<t<1t=0



Bezier CurvesBezier Curves

Bezier curves can be thought of as a higher order extension of linear interpolation

p0

p1

p0

p1p2

p0

p1

p2

p3

Linear Quadratic Cubic



Bezier Curve FormulationBezier Curve Formulation

There are lots of ways to formulate Bezier curves mathematically. Some of these include: de Castlejau (recursive linear interpolations) Bernstein polynomials (functions that define the influence of each control

point as a function of t) Cubic equations (general cubic equation of t) Matrix form

We will briefly examine each of these



Bezier CurveBezier Curve

p0

p1

p2

p3

Find the point x on the curve as a function of parameter t:

x(t)•



de Casteljau Algorithmde Casteljau Algorithm

The de Casteljau algorithm describes the curve as a recursive series of linear interpolations

This form is useful for providing an intuitive understanding of the geometry involved, but it is not the most efficient form




p0

p1

p2

p3

We start with our original set of points

In the case of a cubic Bezier curve, we start with four points




p0

q0

p1

p2

p3

q2

q1

322

211

100

,,

,,

,,

ppq

ppq

ppq

tLerp

tLerp

tLerp




q0

q2

q1

r1

r0

211

100

,,

,,

qqr

qqr

tLerp

tLerp




r1x

r0•

10 ,, rrx tLerp



Bezier CurveBezier Curve

x•

p0

p1

p2

p3



Recursive Linear InterpolationRecursive Linear Interpolation

322

211

100

,,

,,

,,

ppq

ppq

ppq

tLerp

tLerp

tLerp

211

100

,,

,,

qqr

qqr

tLerp

tLerp

10 ,, rrx tLerp

3

2

1

0

p

p

p

p

2

1

0

q

q

q

1

0

r

rx

3

2

1

0

p

p

p

p



Expanding the LerpsExpanding the Lerps

3221

211010

3221211

2110100

32322

21211

10100

111

1111,,

111,,

111,,

1,,

1,,

1,,

pppp

pppprrx

ppppqqr

ppppqqr

ppppq

ppppq

ppppq

ttttttt

ttttttttLerp

tttttttLerp

tttttttLerp

tttLerp

tttLerp

tttLerp



Bernstein Polynomial FormBernstein Polynomial Form

3

32

23

123

023

33

22

12

03

3221

2110

33

363133

13131

111

1111

pp

ppx

ppppx

pppp

ppppx

ttt

tttttt

tttttt

ttttttt

ttttttt



Cubic Bernstein PolynomialsCubic Bernstein Polynomials

33

3

2332

2331

2330

3332

321

310

30

33

223

123

023

33

363

133

33

363133

ttB

tttB

ttttB

ttttB

tBtBtBtB

ttt

tttttt

ppppx

pp

ppx



Bernstein PolynomialsBernstein Polynomials

B33B0

3

B13 B2

3




1

!!

!1

122

12

33

363

133

11

10

222

221

220

333

2332

2331

2330

tB

ini

n

i

ntt

i

ntB

ttB

ttB

ttB

tttB

tttB

ttB

tttB

ttttB

ttttB

ni

iinni




n

ii

ni

iinni

tBt

tti

ntB

0

1

px

Bernstein polynomial form of a Bezier curve:




We start with the de Casteljau algorithm, expand out the math, and group it into polynomial functions of t multiplied by points in the control mesh

The generalization of this gives us the Bernstein form of the Bezier curve This gives us further understanding of what is happening in the curve:

We can see the influence of each point in the control mesh as a function of t We see that the basis functions add up to 1, indicating that the Bezier curve is

a convex average of the control points



Cubic Equation FormCubic Equation Form

133

36333

33

363133

010

2210

33210

33

223

123

023

ppp

pppppppx

pp

ppx

t

tt

ttt

tttttt




0

10

210

3210

23

010

2210

33210

33

363

33

133

36333

pd

ppc

pppb

ppppa

dcbax

ppp

pppppppx

ttt

t

tt




If we regroup the equation by terms of exponents of t, we get it in the standard cubic form

This form is very good for fast evaluation, as all of the constant terms (a,b,c,d) can be precomputed

The cubic equation form obscures the input geometry, but there is a one-to-one mapping between the two and so the geometry can always be extracted out of the cubic coefficients



Cubic Matrix FormCubic Matrix Form

0

10

210

3210

23

33

363

33

pd

ppc

pppb

ppppa

dcbax

ttt

3

2

1

0

23

0001

0033

0363

1331

1

p

p

p

p

d

c

b

a

d

c

b

a

x ttt



Cubic Matrix FormCubic Matrix Form

zyx

zyx

zyx

zyx

ppp

ppp

ppp

ppp

ttt

ttt

333

222

111

000

23

3

2

1

0

23

0001

0033

0363

1331

1

0001

0033

0363

1331

1

x

p

p

p

p

x



Matrix FormMatrix Form

Ctx

GBtx

x

BezBez

zyx

zyx

zyx

zyx

ppp

ppp

ppp

ppp

ttt

333

222

111

000

23

0001

0033

0363

1331

1



Matrix FormMatrix Form

We can rewrite the equations in matrix form This gives us a compact notation and shows how different forms

of cubic curves can be related It also is a very efficient form as it can take advantage of existing

4x4 matrix hardware support…



Bezier Curves & Cubic CurvesBezier Curves & Cubic Curves

By adjusting the 4 control points of a cubic Bezier curve, we can represent any cubic curve

Likewise, any cubic curve can be represented uniquely by a cubic Bezier curve There is a one-to-one mapping between the 4 Bezier control points (p0,p1,p2,p3)

and the pure cubic coefficients (a,b,c,d) The Bezier basis matrix BBez (and it’s inverse) perform this mapping There are other common forms of cubic curves that also retain this property

(Hermite, Catmull-Rom, B-Spline)



DerivativesDerivatives

Finding the derivative (tangent) of a curve is easy:

cbax

dcbax ttdt

dttt 23 223

d

c

b

a

x

d

c

b

a

x 01231 223 ttdt

dttt



TangentsTangents

The derivative of a curve represents the tangent vector to the curve at some point

tdt

dx

tx



Convex Hull PropertyConvex Hull Property

If we take all of the control points for a Bezier curve and construct a convex polygon around them, we have the convex hull of the curve

An important property of Bezier curves is that every point on the curve itself will be somewhere within the convex hull of the control points

p0

p1

p2

p3



ContinuityContinuity

A cubic curve defined for t ranging from 0 to 1 will form a single continuous curve and not have any gaps

We say that it has geometric continuity, or C0 continuity We can easily see that the first derivative will be a continuous quadratic function

and the second derivative will be a continuous linear function The third derivative (and all others) are continuous as well, but in a trivial way

(constant), so we generally just say that a cubic curve has second derivative continuity or C2 continuity

In general, the higher the continuity value, the ‘smoother’ the curve will be, although it’s actually a little more complicated than that…



Interpolation / ApproximationInterpolation / Approximation

We say that cubic Bezier curves interpolate the two endpoints (p0 & p3), but only approximate the interior points (p1 & p2)

In geometric design applications, it is often desirable to be able to make a single curve that interpolates through several points



Piecewise CurvesPiecewise Curves

Rather than use a very high degree curve to interpolate a large number of points, it is more common to break the curve up into several simple curves

For example, a large complex curve could be broken into cubic curves, and would therefore be a piecewise cubic curve

For the entire curve to look smooth and continuous, it is necessary to maintain C1 continuity across segments, meaning that the position and tangents must match at the endpoints

For smoother looking curves, it is best to maintain the C2 continuity as well



Connecting Bezier CurvesConnecting Bezier Curves

A simple way to make larger curves is to connect up Bezier curves Consider two Bezier curves defined by p0…p3 and v0…v3

If p3=v0, then they will have C0 continuity If (p3-p2)=(v1-v0), then they will have C1 continuity C2 continuity is more difficult…

p0

p0

p1

p2

P3

P3

p2

p1

v0

v1

v2

v3

v3

v2

v1

v0

C0 continuity C1 continuity

Computing & Information Sciences Kansas State University Lecture 27 of 42CIS 636/736: (Introduction...

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Transcript of Computing & Information Sciences Kansas State University Lecture 27 of 42CIS 636/736: (Introduction...