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7/27/2019 Bspline Curvesb
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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(