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Morphing Rational B-splineCurves and SurfacesUsing Mass Distributions

COMPUTER GRAPHICS

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Morphing Transforms one target shape into another

Vertex Correspondence

Vertex Interpolation Parametric curves and surfaces

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Linear Interpolation Averaging in affine space

Uniform transition

Every point moves at same

speed

Unsatisfactory artifacts

Flattening, wriggles, etc.

QtPtM(t) )1(

t = 0

t = .25

t = .5

t = .75

t = 1

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Weighted Averaging Interpolation using masses and geometric

positions

Influence ofrelative mass

Larger mass has more impact

Different points morph atdifferent speeds

Less flattening and wriggles

QP

QP

mtmt

QmtPmtM(t)

)1(

)1( t = 0

t = .25

t = .5

t = .75

t = 1

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Rational B-splines A rational B-spline curve of degree n

pk

n

kk

p

k

n

kkk

uNw

uNPw

uP0

0

)(

)(

)( Mass

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Local Morph Control Modification of mass distribution changes the

morphing behaviorlocally

Re-formulate rational B-splines to permitassignment of auxiliary mass for morphing

Customizable morphing between fixed targets

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Local Morph Control Modification of mass distribution changes the

morphing behaviorlocally

Re-formulate rational B-splines to permitassignment of auxiliary mass for morphing

Customizable morphing between fixed targets

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Mass Modification Transition curve

Normalized Distance

curve

)()()1(

)(),(

umtumt

umtutD

QP

Q

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

umtumtuQumtuPumtutM

QP

QP

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Customize Morphing Two easy steps (can be repeated)

Select time frame t0

Edit the normalized distance curve (surface)

Real-time Morph editing environment

Fast computation

Calculations only involve simple algebra

Easy to use

User needs no knowledge of B-spline or mass

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Morph Editing GUI

Time (t)

Normalized

Distance

Surface

Control

PointsSelection

Morph

View

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Conclusion Contributions

Smooth, non-uniform morphing of rational B-

spline curves and surfaces Local morph control by modification of the

associated mass distribution

User interface for real-time morph editing with no

knowledge of B-spline required

Applications

Computer Animation

Model design

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Appendix - Mass Point Definition: a non-zero massmattached to a

pointPin affine space.

Notation: mP/m Operations:

Scalarmultiplication

Addition

mc

Pmc

m

Pmc

QP

QP

Q

Q

P

P

mm

QmPm

m

Qm

m

Pm

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Appendix

Auxiliary Masses P(u) can be rewritten as

Where mp(u) is a new mass distribution functiondefined by

Here wkare auxiliary positive masses attached toeach control point ofP(u)

)(

)()()(

um

uPumuP

p

k

n

kk uNwum0

)()(

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Appendix

Compute Mass Normalized distance between two curves P(u)

and Q(u) with auxiliary masses wkand vk

forms a degree n rational B-spline curve withcontrol pointsRkand weights Wk

Conversely, given WkandRkat t, we have

kkk

k

kk vtwtW

W

vtR )1(and

t

RWv

t

RWw kkk

kkk

and

)1(

)1(