cs559-23 (1)

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12/2/2004 University of Wisconsin, CS559 Fall 2004

Last Time

Subdivision

Sphere

Fractals

Modified Butterfly scheme

Implicit Surfaces

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Today

Parametric curves

Hermite curves

Bezier curves

Homework 7 due Thursday Dec 9

Will be available for collection early in exam week

Best 5 of 7 homeworks are used to compute grade, each one equally

weighted

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Subdivision Shortcomings

Subdivision surfaces suffer from parameterization problems

How are texture coordinates handled?

Surface are 2D in nature, how do we attached a 2D space to

subdivision surfaces?

Subdivision surfaces can be difficult to model with

Still may take complex underlying surface to get desired shape

Effects of changes to underlying mesh are not always obvious

Evaluation is non-trivial

Methods exist for taking a point in the underlying mesh and figuringout where it will go on the surface, but it isnt easy

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What are Parametric Curves?

Define a parameter space

1D for curves

2D for surfaces

Define a mapping from parameter space to 3D points

A function that takes parameter values and gives back 3D points

The result is a parametric curve or surface

0 t1

Mapping:F:t(x,y)

0

1(Fx(t), Fy(t))

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Why Parametric Curves?

Parametric curves are intended to provide the generality of

polygon meshes but with fewer parameters for smooth

surfaces

Polygon meshes have as many parameters as there are vertices (at

least)

Fewer parameters makes it faster to create a curve, and

easier to edit an existing curve

Normal vectors and texture coordinates can be easily

defined everywhere

Parametric curves are easier to animate than polygon

meshes

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Parametric Curves

We have seen the parametric form for a line:

Note that x, y and z are each given by an equation that

involves:

The parameter t

Some user specified control points,x0 andx1 This is an example of a parametric curve

10

10

10

)1(

)1(

)1(

tzztz

tyyty

txxtx

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Basis Functions (first sighting)

A line is the sum of two functions multiplied by vectors:

A linear combination ofbasis functions

tand 1-tare the basis functions

The weights are called control points (x0,y0,z0) and (x1,y1,z1) are the control points

They control the shape and position of the curve

1

1

1

0

0

0

1

z

y

x

t

z

y

x

t

z

y

x

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Hermite Spline

A spline is a parametric curve defined by control points

The term spline dates from engineering drawing, where a spline wasa piece of flexible wood used to draw smooth curves

The control points are adjusted by the userto control the shape ofthe curve

AHermite spline is a curve for which the user provides:

The endpoints of the curve

The parametric derivatives of the curve at the endpoints (tangentswith length)

The parametric derivatives are dx/dt, dy/dt, dz/dt That is enough to define a cubic Hermite spline

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Control Point Interpretation

0

x

0

dt

dx1dt

dx

1x

Start Point

End Point

Start Tangent End Tangent

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Hermite Spline (2)

Say the user provides

A cubic spline has degree 3, and is of the form:

For some constants a, b, c and d derived from the control points, but

how?

We have constraints:

The curve must pass throughx0 when t=0

The derivative must be x0 when t=0

The curve must pass throughx1 when t=1

The derivative must be x1 when t=1

dctbtatx

23

1

1

0

0

10 ,,,dt

ddt

d xxxx

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Hermite Spline (3)

Solving for the unknowns gives:

Rearranging gives:0

0

0101

0101

233

22

xd

xc

xxxxb

xxxxa

)2(

)(

)132(

)32(

230

23

1

23

0

23

1

ttt

tt

tt

tt

x

x

x

xx

10121

0011

1032

0032

2

3

0101t

t

t

xxxxxor

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Basis Functions

A point on a Hermite curve is obtained by multiplying each

control point by some function and summing

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

x1

x0

x'1

x'0

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Splines in 2D and 3D

For higher dimensions, define the control points in higher

dimensions (that is, as vectors)

10121

0011

1032

0032

2

3

0101

0101

0101

t

t

t

zzzz

yyyyxxxx

z

yx

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Bezier Curves (1)

Different choices of basis functions give different curves

Choice of basis determines how the control points influence the

curve

In Hermite case, two control points define endpoints, and two more

define parametric derivatives

For Bezier curves, two control points define endpoints, and

two control the tangents at the endpoints in a geometric way

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Control Point Interpretation

0

x

3x

Start Point

End Point

Point along start tangent

Point along end Tangent

2x

1x

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Bezier Curves (2)

The user supplies dcontrol points, pi

Write the curve as:

The functionsBidare theBernstein polynomials of degree d

Where else have you seen them?

This equation can be written as a matrix equation also

There is a matrix to take Hermite control points to Bezier controlpoints

d

i

d

ii tBt

0

px ididi tti

dtB

1

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Bezier Basis Functions for d=3

0

0.2

0.4

0.6

0.8

1

1.2

B0B1

B2

B3

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Bezier Curves of Varying Degree

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Bezier Curve Properties

The first and last control points are interpolated

The tangent to the curve at the first control point is along theline joining the first and second control points

The tangent at the last control point is along the line joining

the second last and last control points The curve lies entirely within the convex hull of its control

points The Bernstein polynomials (the basis functions) sum to 1 and are

everywhere positive

They can be rendered in many ways

E.g.: Convert to line segments with a subdivision algorithm

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Rendering Bezier Curves (1)

Evaluate the curve at a fixed set of parameter

values and join the points with straight lines

Advantage: Very simple

Disadvantages: Expensive to evaluate the curve at many points

No easy way of knowing how fine to sample

points, and maybe sampling rate must be different

along curve

No easy way to adapt. In particular, it is hard tomeasure the deviation of a line segment from the

exact curve

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Rendering Bezier Curves (2)

Recall that a Bezier curve lies entirely within the convex

hull of its control vertices

If the control vertices are nearly collinear, then the convex

hull is a good approximation to the curve

Also, a cubic Bezier curve can be subdividedinto two

shorter curves that exactly cover the original

This suggests an algorithm:

Keep breaking the curve into sub-curves

Stop when the control points of each sub-curve are nearly collinear

Draw the control polygon - the polygon formed by the control points

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Invariance

Translational invariance means that translating the control points and

then evaluating the curve is the same as evaluating and then translating

the curve

Rotational invariance means that rotating the control points and then

evaluating the curve is the same as evaluating and then rotating the

curve

These properties are essential for parametric curves used in graphics

It is easy to prove that Bezier curves, Hermite curves and everything

else we will study are translation and rotation invariant

Some forms of curves, rational splines, are alsoperspective invariant Can do perspective transform of control points and then evaluate the curve

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Longer Curves

A single cubic Bezier or Hermite curve can only capture a small class ofcurves

At most 2 inflection points

One solution is to raise the degree

Allows more control, at the expense of more control points and higher

degree polynomials Control is not local, one control point influences entire curve

Alternate, most common solution is to join pieces of cubic curvetogether intopiecewise cubic curves

Total curve can be broken into pieces, each of which is cubic

Local control: Each control point only influences a limited part of the curve Interaction and design is much easier

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Piecewise Bezier Curve

knot

P0,0

P0,1 P0,2

P0,3

P1,0

P1,1

P1,2

P1,3

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Continuity

When two curves are joined, we typically want some degree

of continuity across the boundary (the knot)

C0, C-zero, point-wise continuous, curves share the same point

where they join

C1, C-one, continuous derivatives, curves share the sameparametric derivatives where they join

C2, C-two, continuous second derivatives, curves share the same

parametric second derivatives where they join

Higher orders possible

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Bezier Continuity

P0,0

P0,1 P0,2

J

P1,1

P1,2

P1,3

Disclaimer: PowerPoint curves are not Bezier curves, they areinterpolating piecewise quadratic curves! This diagram is an

approximation.

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Sketch of Proof for C1

3

3

32

2

32

1

32

0

3

3

2

2

2

1

3

0

)(3)2(3)331(

)1(3)1(3)1(

ttttttttt

tttttt

xxxx

xxxxx

2

3

2

2

2

1

2

0 3)32(3)341(3)363( tttttttdt

dxxxx

x

Bezier curve equation:

Parametric derivative:

)(333 23321

xxxxx

tdt

d)(333 0110

0

xxxxx

tdt

d

Evaluated at endpoint of curve (note proves tangent property):

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Proof (cont)

P0,0

P0,1 P0,2

J

P1,1 P1,2

P1,3

2,01,1

2,01,1 )(3)(3

PJJP

PJJP

C1 requires equal parametric derivatives:

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Geometric Continuity

Derivative continuity is important for animation

If an object moves along the curve with constant parametric speed, thereshould be no sudden jump at the knots

For other applications, tangent continuity might be enough

Requires that the tangents point in the same direction

Referred to as G1 geometric continuity Curves couldbe made C1 with a re-parameterization: u=f(t)

The geometric version ofC2 is G2, based on curves having the same radiusof curvature across the knot

What is the tangent continuity constraint for a Bezier curve?

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Bezier Geometric Continuity

P0,0

P0,1 P0,2

J

P1,1 P1,2

P1,3

)()( 2,01,1 PJJP k for some k

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Parametric Surfaces

Define points on the surface in terms of two parameters

Simplest case: bilinear interpolation

s

t

s

x(s,t)

P0,0

P1,0

P1,1P0,1

x(s,0)

x(s,1)

1

0

1

0

,,,

,1,0

,1,0

1,11,0

0,10,0

)()(),(

,1

,1

)1,()0,()1(),(

)1()1,(

)1()0,(

i j

tjsiji

tt

ss

tFsFPtsx

tFtF

sFsF

stxsxttsx

sPPssx

sPPssx

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Tensor Product Surface Patches

Defined over a rectangular domain

Valid parameter values come from within a rectangular region in

parameter space: 0s

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Bezier Patches

As with Bezier curves,Bin(s) and

Bjm

(t) are the Bernstein polynomialsof degree n and m respectively

Most frequently, use n=m=3: cubicBezier patch

Need 4x4=16 control points, Pi,j

n

i

m

j

m

j

n

iji tBsBts0 0

,, Px

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12/2/2004 University of Wisconsin CS559 Fall 2004

Bezier Patches (2)

Edge curves are Bezier curves

Any curve of constant s or tis a Bezier curve

One way to think about it:

Each row of 4 control points defines a Bezier curve in s

Evaluating each of these curves at the same s provides 4 virtual control

points The virtual control points define a Bezier curve in t

Evaluating this curve at tgives the point x(s,t)

x(s,t)

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cs559-23 (1)