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Agenda

Parametric curves in space

Parametric surfaces in space

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Parametric Curves

The idea is to be able to define complexcurves (or surfaces in 3D) by controlling a

set of points. This can be used for:

Designing objects

Designing trajectories of objects or cameras inanimations

Design surfaces

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Parametric Curves

See examples in Paint:

You define a starting and ending point.

Then you define two additional controlpoints The curve passes through the first and lastcontrol points

The curve gets close to the intermediatecontrol points

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Bzier Curves

Invented by French engineer Pierre Bzier

Are simple to implement

Have several features that are useful indesigning objects

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Bzier Curves

Desirable features Order-0 continuity: curve n+1 starts where

curve n ends. Order-1 continuity: the derivative of curve n+1where it starts is the same as the derivative ofcurve n where it ends.

The curve is confined within the convex-hull formed by the control points

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Bzier Curves

Some examples:

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Bzier Curves

How to create a closed curve:

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Bzier Curves

How to make the curve pass closer to apoint:

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Bzier Curves the Math

For n+1 controlpoints:

BEZ, the blendingfunctions, are theBernsteinpolynomials:

C(n,k) are thebinomialcoefficients:

n

k

nkk uuBEZu0

,10),()( pP

1

, )1(),()(

kk

nk uuknCuBEZ

)!(!

!),(

knk

nknC

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Bzier Curves the Math

The vectorialequationrepresents three

parametricequations:

n

k

nkk uBEZxux0

, )()(

n

k

nkk uBEZyuy0

, )()(

n

k

nkk uBEZzuz0

,)()(

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Bzier Curves the Math

For instance, for n = 3

3

3,0 )1()( uuBEZ 2

3,1 )1(3)( uuuBEZ

)1(3)(2

3,2 uuuBEZ

3

3,3 )( uuBEZ

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Bzier Curves the Math

Blending functions

from: www.cs.virginia.edu/~gfx/Courses/2001/Intro.fall.01/Lectures/lecture21.ppt

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Bzier Curves

Zero and one continuity The curve connects the first and last control

points:

Values for the parametric first derivatives of aBzier curve can be calculated as:

np

p

)1(

)0( 0

P

P

nn nn

nn

ppP

ppP

1

10

)1('

)0('

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Bzier Curves

Therefore, it is easy to connect two curveswith 0 and 1 continuity:

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Parametric Surfaces

Parametric Bzier surfaces:

Note that two (u, v) parameters areneeded.

n

k

nkmjkj

m

j

uBEZvBEZpvuP0

,,,

0

,

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Parametric Surfaces

Examples:

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Parametric Surfaces

Surfaces can also be connected with 0-and 1-continuity:

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Other types of curves (andsurfaces

Hermite Interpolation

Cardinal Splines

Kochanek-Bartels Splines B-splines

Non-uniform rational b-spline curves

(NURBS)