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Graphics II 91.547

Parametric Curvesand Surfaces

Session 3

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Parametric Curves and Surfaces:Key Concepts

Point

1

z

y

x

Curve, Surface Collection of points

Parametric Representation 1...0

1

)(

)(

)(

uuz

uy

ux

(for a curve)

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Parametric Curves and SurfacesKey Concepts, contd.

Multiplying a point by a scalar: Scaling the representation

Polynomial

33

2210

0

)(

)(

ucucuccux

uu

xxxx

n

k

kk

cp

Control Points Set of (degree+1) points that bear somerelation to the desired curve or surface

Basis FunctionsSet of (degree + 1) polynomials that, whenmultiplied by the corresponding controlpoints, produce the curve or surface

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Continuity Considerations: C0 & C1 Continuity

zyx du

udz

du

udy

du

udx

du

udiii

p )()()()(

p( )u

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How do we insure C0 and C1 continuity?

p1

p0

p2

p3

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Bezier Curves:

1. Require interpolation at the endpoints of the curve:

p p

p po

( )

( )

0

13

2. Require that the line segments P0 P1 and P3P2 match the first derivative of the curve at 0 and 1.

p1

p0

p2

p3

pp p

p p

pp p

p p

'( ) ( )

'( ) ( )

0 3

1 3

1 0

13

1 0

3 2

13

3 2

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Calculating the Bezier Geometry Matrix

p c c c c c

p c

p c c c c

p p c

p p c c c

( )

( )

u u u u ukk

k

0 1 2

2

3

3

0

3

0 0

3 0 1 2 3

1 013 1

3 213 1 2 32 3

Inserting the four conditions gives:

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Calculating the Bezier Geometry Matrix

This gives four equations in four unknowns that

can be solved for :

where is the Bezier Geometry Matrix

c M p

M

M

B

B

B

1 0 0 0

3 3 0 0

3 6 3 0

1 3 3 1

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Bezier Blending Functions

p u M p

p b p

b M u

( )

( ) ( )

( )

u

u u

u

u u u

u u u

u u

u

T

B

T

B

T

1 3 3

3 6 3

3 3

2 3

2 3

2 3

3

where

This can be expressed:

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Bezier Blending Functions:

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Convex Hull Property:

p1

p0

p2

p3

b uii

0

3

1( )

Curve must lie withinthis region known as“convex hull” of thecontrol points.

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Further look at the Bezier blending functions:

b( )

( )

( )

( )

( )!

!( )!( )

u

u

u u

u u

u

b ud

k d ku ukd

k d k

1

3 1

3 1

1

3

2

2

3

These are an instance of the Bernstein Polynomialsfor k=3.

Factoring the blending functions gives:

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Bezier Curves:Code Example

120 goto 120

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An interesting geometric construction

p0

0

p1

0p2

0

p3

0

p0

1

p1

1

p2

1

p0

2

p1

2

p0

3

p p p

p p

i

r

i

r

i

r

i i

u u u u ur ni n ru

( ) ( ) ( ) ( ),...,,...,

( )

1

10

1

1

1

0

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Recursive subdivision of Bezier curves

p0

0

p1

0p2

0

p3

0

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Bezier surface patch

p00

p30

p03

p33

p p u M PM v( , ) ( ) ( )u v b u b vij o

j

i

i

j ij

T

B B

T

3

0

3

Convex hull of patch

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Bezier surface patchCorner boundary conditions

p p

pp p

pp p

pp p p p

( , )

( , ) ( )

( , ) ( )

( , ) ( )

0 0

0 0 3

0 0 3

0 0 9

00

10 00

01 00

2

00 01 10 11

u

v

u v

Four equations for each corner gives 16 total.

Patch must interpolate the corner.

Defines derivative in u direction.

Defines derivative in v direction.

“Twist”

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Rendering the Bezier patch by recursivesubdivision

First subdivide curves ofconstant v.

Connect new controlpoints to form newcurves.

Finally subdivide these curves to form 4 new patches.

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Recursive subdivision of Bezier curves

p0

0

p1

0p2

0

p3

0

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The Utah Teapot: 32 Bezier Patches

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Utah Teapot: Polygon Representation

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Rendering the Teapot

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Conversion between curve representations

Given a cubic Bezier curve expressed in terms of theBezier geometry matrix, MB:

The same curve can be expressed in terms of anotherpolynomial basis with matrix M and matrix of control

points q.

Solving for q gives:

q M M p 1

B

p u Mq( )u T

p u M p( )u T

B

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Conversion between curve representations:an example

Suppose you have a matrix of four control points, q and

a cubic interpolating those points q(u). The curve is obtained

by the interpolating geometry matrix MI :

where

We want to find p such that:

p M M q M M

I B I B

1 156

32

13

13

32

56

1 0 0 0

3

3

0 0 0 1

where

q u M q( )u T

I M AI 1

q u M p( )u T

B

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Drawing Bezier curves and surfaces:Evaluators

Evaluator

ControlPoints

ParameterGeneration

glMap1{fd}()glMap2{fd}()

glEvalCoord1{fd}()glEvalCoord2{fd}()

Parameter(s)

Curve/surfacevalue

Establish Evaluator:

Evaluate Points:

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One Dimensional Evaluators

Void glMap1{fd](GLenum target, TYPE u1, TYPE u2, Glint stride,Glint order, TYPE *points);

target: specifies what the control points represent

u1: beginning parameter valueu2: end parameter valuestride: number of values in each block of

control point storageorder: order of the Bezier curve (degree+1)*points: pointer to array of control points

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One Dimensional Evaluators:target parameter

GL_MAP_VERTEX_3 x,y,z vertex coords.GL_MAP_VERTEX_4 x,y,z,w vertex coordsGL_MAP1_INDEX color indexGL_MAP1_COLOR_4 RGBAGL_MAP2_NORMAL normal coordinatesGL_MAP1_TEXTURE_COORD_1 s texture coords.GL_MAP1_TEXTURE_COORD_2 s,t texture coords.GL_MAP1_TEXTURE_COORD_3 s,t,r texture coords.GL_MAP1_TEXTURE_COORD_4 s,t,r,q texture coords.

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One Dimensional Evaluators:Code Example

120 goto 120

?

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Mesh Generation:One Dimensional

void glMapGrid1{fd} (Glint n, TYPE u1, TYPE u2);

Defines a grid that goes from u1 to u2 in n evenly spaced steps.

void glEvalMesh1 (GLenum mode, GLint p1, Glint p2);

mode is either GL_POINT or GL_LINE depending on whether you want to draw individual points or connected lines.

p1, p2 are the grid values between which the mesh is to beevaluated

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Two Dimensional Evaluators

Void glMap2{fd](GLenum target, TYPE u1, TYPE u2, Glint ustride,Glint uorder, TYPE v1, TYPE v2, GLint vstride, Glint vorder, TYPE *points);

target: specifies what the control points represent

u1, v1: beginning parameter valueu2, v2: end parameter valuestride: number of values in each block of

control point storageorder: order of the Bezier curve (degree+1)*points: pointer to array of control points

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Mesh Generation:Two Dimensional

void glMapGrid2{fd} (GLint nu, TYPE u1, TYPE u2 GLint nv, TYPE v1, TYPE v2);

Defines a two dimensional grid that goes from u1 to u2 in nu evenly spaced steps and from v1 to v2 in nv evenly spaced steps.

void glEvalMesh2 (GLenum mode, GLint p1, GLint p2, GLint q1, GLint q2);

mode is either GL_POINT , GL_LINE , or GL_FILL depending on whether you want to draw individual points, connected lines,or a filled mesh.

p1, p2, q1, q2 are the grid values between which the meshis to be evaluated

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Two Dimensional Evaluators:Code Examples

120 goto 120