# Morphing Rational spline Curves and Surfaces Using Mass by sanmelody

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```									Morphing Rational B-spline
Curves and Surfaces
Using Mass Distributions

Tao Ju, Ron Goldman
Department of Computer Science
Rice University
Morphing
   Transforms one target shape into another
   Vertex Correspondence
   Vertex Interpolation
   Parametric curves and surfaces
Linear Interpolation
   Averaging in affine space
t=0

M(t)  (1  t ) P  t Q          t = .25

   Uniform transition               t = .5
   Every point moves at same
speed                        t = .75
   Unsatisfactory artifacts         t=1
   Flattening, wriggles, etc.
Weighted Averaging
   Interpolation using masses and geometric
positions
t=0
(1  t ) mP P  t mQQ
M(t) 
(1  t ) mP  t mQ          t = .25

   Influence of relative mass            t = .5
   Larger mass has more impact
   Different points morph at         t = .75
different speeds
   Less flattening and wriggles      t=1
Rational B-splines
   A rational B-spline curve of degree n
k 0
p
wk Pk N kn (u )
P(u ) 
k 0
p
wk N kn (u )     Mass
Linear vs. Weighted Averaging
Local Morph Control
   Modification of mass distribution changes the
morphing behavior locally
   Re-formulate rational B-splines to permit
assignment of auxiliary mass for morphing
   Customizable morphing between fixed targets
Local Morph Control
   Modification of mass distribution changes the
morphing behavior locally
   Re-formulate rational B-splines to permit
assignment of auxiliary mass for morphing
   Customizable morphing between fixed targets
Mass Modification
   Transition curve

(1  t )mP (u) P(u)  t mQ (u) Q(u)
M (t , u) 
(1  t )mP (u)  t mQ (u)

  Normalized Distance
curve
t mQ (u)
D(t , u ) 
(1  t )mP (u)  t mQ (u )
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
Morph Editing GUI

Control
Points
Selection
Morph
View
Normalized
Distance
Surface

Time (t)
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
Appendix - Mass Point
   Definition: a non-zero mass m attached to a
point P in affine space.
   Notation: mP/m
   Operations:
m P cm P
   Scalar multiplication      c    
m    cm

mP P m QQ mP P  mQQ
mP    mQ   mP  mQ
Appendix – Auxiliary Masses
   P(u) can be rewritten as
m (u )  P(u )
P (u ) 
m (u )
   Where mp(u) is a new mass distribution function
defined by
p
m (u )   wk N kn (u )
k 0

   Here wk are auxiliary positive masses attached to
each control point of P(u)
Appendix – Compute Mass
   Normalized distance between two curves P(u)
and Q(u) with auxiliary masses wk and vk
forms a degree n rational B-spline curve with
control points Rk and weights Wk
t vk
Rk       and Wk  (1  t ) wk  t vk
Wk
   Conversely, given Wk and Rk at t, we have
Wk (1  Rk )            Wk Rk
wk                 and vk 
(1  t )                t

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