Representation of Curves and Surfaces

COLLEGE OF ENGINEERING UNIVERSITY OF PORTO

COMPUTER GRAPHICS AND INTERFACES /GRAPHICS SYSTEMS JGB / AAS 2004

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Representation of Curves and Surfaces

Graphics Systems /Computer Graphics and Interfaces



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Representation of Curves and SurfacesRepresentation of surfaces: Allow describing objects through their cheeks.

The three most common representations are:– Polygonal mesh– Bicúbicas parametric surfaces– Quadratic surfaces

Parametric representation of curves: Important in computer graphics and 2D because of parametric surfaces are a generalization of these curves.

"Tea-pot" modeled by smooth curved surfaces (bicúbicas).

Reference model in computer graphics, especially for new techniques of realism texture and surface testing.Created by Martin Newell (1975)



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Polygonal Mesh

Polygonal Mesh: Is a collection of edges, vertices and polygons interconnected so that each edge is shared only a maximum of two polygons.

Curve polylineSection of a curved object.The approximation error can be reduced by increasing the number of polygons.

3D object represented by polygon mesh.



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Polygonal Mesh

Characteristics of polygonal mesh:– An edge connects two vertices.– A polygon is defined by a closed sequence of edges.– An edge is shared by two adjacent polygons or 1.– A vertex is shared by at least two edges.– All edges are part of a polygon.

The data structure for represent the polygonal mesh can have multiple configurations, which are evaluated by memory space and Processing time needed to get a response, for example:

– Get all the edges that join a given vertex.– Determine the polygons that share an edge or a vertex.– Determine the vertices attached to an edge.– Determine the edges of a polygon.– Plot the mesh.– Identify errors in the representation, as the lack of an edge, vertex or polygon.



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Polygonal Mesh

. 1 Representation Explains: each polygon is represented as a list of coordinates of the vertices that constitute it.

An edge is defined by two consecutive vertices and between the last and first on the list.

(X1, y1, z1)

(X2, y2, z2) (X3, y3, z3)

(X4, y4, z4)

P = ((x1, y1, z1), (x2, y2, z2), ..., (xn, yn, zn))

Evaluation of the data structure:– Memory consumption (repeated vertices).– There is an explicit representation of edges

and vertices shared.– In the graphical representation of the same

edge is "clipped"Drawn and more than once.– When you drag a vertex is necessary to

know all the edges that share that vertex.

4x

3x



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Polygonal Mesh

. 2 Representation of Pointers to List of Vertices: each polygon is represented by a list of indices (or pointers) for a list of vertices.

V = ((x1, y1, z1), (x2, y2, z2), ..., (xn, yn, zn))List of Vertices

Advantages:– Each vertex of the polygon mesh is stored only once in memory.– The coordinate of a vertex is easily changed.

Disadvantages:– Hard to get the polygons that share a given edge.– The edges remain "clipped " and drawn once more.

V = (V1, V2, V3, V4) = (x1, y1, z1), (x2, y2, z2), ..., (x4, y4, z4))

P1 = (1,2,4)

P2 = (4,2,3)



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Polygonal Mesh

. 3 Representation of Pointers to List of Edges: each polygon is represented by a list of pointers to a list of edges, wherein each edge appears only once. In turn, each edge points to two vertices that define it and also stores the polygons which it belongs.

A polygon is represented by P = (E1, E2, ..., En) and an edge as E = (V1, V2, P1, P2). If the edge belongs to only one polygon then P2 it is null.



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Polygonal MeshAdvantages:

– The graphic design is easily obtained by scrolling through the list of edges. Not repeating occurs clipping or drawing.

– To fill (color) of the polygons works with a list of polygons. Easy operation is performed clipping on the polygons.

Disadvantages:– Still not immediately determine the edges that focus on the same vertex.

Solution Baumgart

• Each vertex has a pointer to one of the edges (random) which focuses this vertex.

• Each edge has pointers to the edges that cover a vertex.



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Curves Cubic

Motivation: Smooth curves represent the real world.

• Representation by polygonal mesh is a first order approximation:– The curve is approximated by a sequence of linear segments.– Large amount of data (vertices) to obtain the curve precisely.– Unwieldy to change the shape of the curve, ie several points need to

position accurately.

• Are generally used polynomials grade 3 (Curves Cubic), and the complete curve formed by a number of cubic curves.

– degree <3 offer little flexibility in controlling the shape of the curves and for an interpolation between two points with the definition of the derivative at the end points. A polynomial of degree 2 is specified by three points that define the plane where the curve takes place.

– degree > 3 may introduce unwanted oscillations and requires more computational calculation.



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Curves Cubic

The representation of the curves is in the form PARAMETRIC:

x = fx(T), y = fy(T) ex: x = 3t3 + T2 y = 2t3+ T

The explicit form: y = f (x) and x: y = x3+2 X2

1. We can not have multiple values y for the same x

2. Curves can not describe with vertical tangents

The implicit form: f (x, y) = 0 and x: x2+ Y2-R2= 0

1. Restrictions need to be able to model only part of the curve

2. Difficult join two curves smoothly



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Curves Parametric cubic

The figure shows a curve formed by two parametric cubic curves in 2D.



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General representation of the curve:

x (t) = axt3+ Bxt2+ Cxt + dx

y (t) = ayt3+ Byt2+ Cyt + dy

z (t) = azt3+ Bzt2+ Czt + dz0≤ t ≤ 1

Being:

zyx

zyx

zyx

zyx

ddd

ccc

bbb

aaa

C

CTtztytxtQ .)()()()(

123 tttT



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The above representation is used to represent a single curve. Bring together the various segments of the curve?

We intend joining a point geometric continuity and

That have the same slope at the junction smoothness (continuity of the derivative).

Ensuring continuity and smoothness at the junction is ensured by matching the derivatives (tangent) curves at the junction point. To this end calculates:

t

TC

t

CT

t

tz

t

ty

t

tx

t

tQ

)()()()()(

0123 2 ttt

T

With:



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Types of Continuity:

G0 - Zero geometric continuity curves join at a point.

G1 - Geometric continuity is an the direction of the tangent vectors is equal.

C1 - Parametric continuity 1 the tangent at the point of junction have the same direction and amplitude (the first derivative equal).

Cn - Parametric continuity n curves have at the junction point all the same derivatives up to order n.



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If we consider tas timeThe continuity C1 means that the speed of an object moving along the curve remains continuous

The continuity C2 imply that the acceleration would also be continuous.

At the junction point of the curve S the curves C0,C1 and C2 we have:

Continuity G0 between SandC0Continuity C1 between SandC1Continuity C2 between SandC2



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Parametric continuity is more restrictive than the geometric continuity:

For example: C1 implies G1

At the junction point P2 we have:

Q2 and Q3 are G1 with Q1

Only Q2 C is1 with Q1 (TV1= TV2)



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Curves Cubic Parametric - Types of Curves

1. Hermite curves− Continuity G1 at junction points− Geometric vectors:

• 2 endpoints and• The tangent vectors at those points

2. Bezier curves− Continuity G1 at junction points− Geometric vectors:

• 2 endpoints and • 2 points that control the tangent vectors such extremes

3. Curves Splines− Very extended family of curves− Greater control continuity at junction points (C Continuity1

and C2)

P4

P1R4

R1 P3

P4

P1P2



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Common notation

Matrix T

CTtztytxtQ .)()()()(

GMTtQ ..)(

123 ttt

44434241

34333231

24232221

14131211

mmmm

mmmm

mmmm

mmmm

4

3

2

1

G

G

G

G

Matrix Base Matrix Geometric

Matrix Base:Characterizes the type of curve

Matrix Geometric:A given geometrically conditions and contains curve related to the values of the curve geometry.



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Common notation

Conclusion 1: Q (t) is a weighted sum of the elements of the geometric vector

GMTtQ ..)(

4

3

2

1

44434241

34333231

24232221

14131211

23 ..1)(

G

G

G

G

mmmm

mmmm

mmmm

mmmm

ttttQ

44434242

143

34333232

133

2232222

123

14131212

113

).().(

).().()(

GmtmmtmtGmtmmtmt

GmtmmtmtGmtmmtmttQ

Conclusion 2: Weights are cubic polynomials in t FUNCTIONS MIX

GBGMTCTtQ ....)( (Blending functions)



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Hermite curves

HHHHHH GBGMtttGMTtQ ...1..)( 23

HH GMtttQ ..0123)(' 2

Geometric vectors:

4

1

4

1

R

R

P

P

GH

1..1000)0( PGMQ HH

4..1111)1( PGMQ HH

1..0100)0(' RGMQ HH

4..0123)1(' RGMQ HH

HHH G

R

R

P

P

GM

4

1

2

1

..

0123

0100

1111

1000

0001

0100

1233

1122

0123

0100

1111

10001

HM



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Hermite curvesMixing functions (Blending functions)

Q (t) = (2t3-3t2+1) P1 +

(-2t3+3 T2) P4 +

(T3-2t2+ T) R1 +

(T3-T2) R4

HHHHHH GBGMtttGMTtQ ...1..)( 23

0001

0100

1233

1122

HM

4

1

4

1

R

R

P

P

GH

Mixing functions of Hermite curves, referenced by the element of the geometric vector that multiplies, respectively.



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Hermite curves - Example

Left: Functions of heavy mixture by the appropriate factor from the geometric vector.

Center: Y (t) = sum of the four functions of the left

Right: Hermite curve



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Hermite curves - Examples

- P1 and P4 fixed

- R4 fixed

- R1 varies in amplitude

- P1 and P4 fixed

- R4 fixed

- R1 varies in the direction



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Hermite curvesExample of Interactive Design

• The extreme points can be repositioned

• The tangent vectors can be changed by pulling the arrows

• The tangent vectors are forced to be collinear (L continuity1) And R4 is displayed in the opposite direction (higher visibility)

• It is common to have commands to force continuity G0, G1 or C1

Continuity at the junction:

4

1

4

1

R

R

P

P

7

4

7

4

.

R

RK

P

P

• K> 0 G1

• K = 1 C1



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Hermite curves



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Bezier curves

Vector Geometric:

P3

P4

P1P2

4

3

2

1

P

P

P

P

GB

For a given curve demonstrates that compared with GH:

R1 = Q '(0) = 3. (P2- P1)

R4 = Q '(1) = 3. (P4- P3)

4

3

2

1

4

1

4

1

.

3300

0033

1000

0001

P

P

P

P

R

R

P

P

GH

GH M =HB . GB



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Bezier curves

• The same Bezier curve representation: Q (t) = t. MB . GB

BHBHBHBHHH GMMTGMMTGMTtQ ........)(

0001

0033

0363

1331

. HBHB MMM

Q (t) = (1-t)3P1 +

3t (1-t)2P2 +

3t2(1-t) P3 +

t3P4



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Bezier curves

Additional information about the functions of Mixture: - For t = 0 Q (t) = P1 For t = 1 Q (t) = P4 The curve passes P1 and P4

- The sum at any point is 1.

- It is noted that Q (t) is a weighted average of the four control points, Then the curve is contained within the polygon convex (2D) or convex polyhedron (3D) defined by these points, called the "convex hull".

What advantage can we draw here?

Q (t) = (1-t)3P1 +

3t (1-t)2 P2 +

3t2(1-t) P3 +

t3P4



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Bezier curvesBinding of Bezier curvesContinuity G1:

P4 - P3 K = (P5 - P4) With K> 0 i.e. P3P4 and P5 must be collinear

Continuity C1:

P4 - P3 K = (P5 - P4) Restricting K = 1



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Drawing of Cubic curves

Two algorithms:

1. Evaluation x (t), y (t) and z (t) incremental values for t between 0 and 1.

2 Subdivision of the curve:. Casteljau Algorithm

1. Evaluation x (t), y (t) and z (t)

Horner's rule reduces the number of multiplications, 11 additions and 10 to 9 and 10, respectively operations.

f (t) = at3+ Bt2+ Ct + d = ((at + b). + T c). T + d

2. Casteljau algorithm

Perform the recursive subdivision of the curve, stopping only when the curve in question is sufficiently "flat" to be able to be approximated by a line segment.

Efficient Algorithm: 6 requires only shifts and 6 additions in each division.



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Drawing of Cubic curves - Casteljau Algorithm

Possible stopping criteria:

- The curve in question is sufficiently "flat" to be able to be approximated by a line segment.

- The four control points are on the same pixel.

L2 = (P1+ P2) / 2, H = (P2+ P3) / 2, L3 = (L2+ M) / 2, R3 = (P3+ P4) / 2

R2 = (H + R3) / 2, L4 = R1 = (L3 + R2) / 2



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Drawing of Cubic curves

Algorithm Calteljau

void DrawCurveRecSub (curve,ε)

{if (Straight (curve,ε))

DrawLine (curve);

else {

SubdivideCurve (curve,leftCurve, rightCurve);

DrawCurveRecSub (leftCurve, ε);

DrawCurveRecSub (rightCurve, ε);

}

}



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Exercise



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Cubic Surfaces

Cubic surfaces are a generalization of cubic curves. The equation of the surface is obtained from the equation of the curve:

Q (t) = T. M. G, being G constant.

Switch to variable s:Q (s) = S. M. G

By varying the points of the vector G3D eométrico along a path parameterized by t are obtained:

)(

)(

)(

)(

..)(..),(

4

3

2

1

tG

tG

tG

tG

MStGMStsQThe geometrical matrix is composed of 16 points.



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Surface Hermite

For the x coordinate:

x

HHxH

tR

tR

tP

tP

MStGMStsx

)(

)(

)(

)(

..)(..),(

4

1

4

1

x

Hx

g

g

g

g

MTtP

14

13

12

11

1 ..)(

x

Hx

g

g

g

g

MTtP

24

23

22

21

4 ..)(

x

Hx

g

g

g

g

MTtR

34

33

32

31

1 ..)(

x

Hx

g

g

g

g

MTtR

44

43

42

41

4 ..)(

TTHHx

TTH

x

TMGTM

gggg

gggg

gggg

gggg

tR

tR

tP

tP

....

)(

)(

)(

)(

24434241

24333231

24232221

14131211

4

1

4

1

It is concluded that: TTHHxH TMGMStsx ....),(

HxG



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Surface Hermite



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Bezier Surface

The equations for the Bezier surface can be obtained in the same way that the Hermite, resulting in:

TTBBxB TMGMStsx ....),(

TTBByB TMGMStsy ....),(

TTBBzB TMGMStsz ....),(

The geometric matrix has 16 control points.



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Bezier Surface

Continuity C0 and G0 is obtained by matching the four points of border control: P14P24P34P44

For G1 should be collinear:

P13P14 and P15

P23P24 and P25

P33P34 and P35

P43P44 and P45

and

(P14-P13) / (P15 - P14) = K

(P24-P23) / (P25 - P24) = K

(P34-P33) / (P35 - P34) = K

(P44-P43) / (P45 - P44) = K

Representation of Curves and Surfaces

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