Transcript PowerPoint

Cubic Surfaces
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CGP&P Chapter 11
Modeling surfaces
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Extension of parametric cubic curves
called “parametric bicubic surfaces”
Idea: infinite # of curves stacked
together
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equations now have 2 parameters
Q(s,t)
P (t)
t=0.75
1
P4(t)
t=0.25
t
s
Matrix representation
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A single curve was expressed
Q(t) = TMG
Now, the geometric information varies
Q(s,t) = SMG
 G1 (t ) 
G (t )
2


G
G3 (t ) 


G4 (t )
each Gi(t) is itself
a cubic curve
Matrix representation (cont.)
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Since each Gi is a cubic curve, it can be
written:
Gi(t) = TMGi
 g i1 
g 
i2 

Gi 
 gi3 
 
 gi 4 
Matrix representation (cont.)
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Substituting, we obtain:
TMG1 
TMG 
2
Q(s,t) = SM 
TMG3 


TMG4 
 G1 
G 
 2  This formulation does not work in
= SMTM G3  terms of matrix dimensions...
 
G4 
Matrix representation (cont.)
So, use the transpose rule:
Gi(t) = GiT MT TT
 G1T 
 T T T
Q(s,t) = S M G2  M T
G T 
3
 T
G4 
=SM
 g11
g
 21
 g 31

 g 41
g12
g 22
g13
g 23
g 32
g 42
g 33
g 43
g14 
g 24 
g 34 

g 44 
MT TT
Matrix representation (cont.)
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Finally, remember that the large
geometry matrix has 3 components (x,
y, z) for each gij, so that we get three
parametric equations:
x(s,t) = S M Gx MT TT
y(s,t) = S M Gy MT TT
z(s,t) = S M Gz MT TT
Hermite surfaces
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extension of Hermite curves to
parametric bicubic surfaces
four elements of the geometry matrix
are now P1(t), P4(t), R1(t), R4(t)
can be thought of as interpolating the
curves Q(s,0) and Q(s,1) or
Q(0,t) and Q(1,t)
Hermite surface matrices
x(s,t) = S M GHx MT TT
y(s,t) = S M GHy MT TT
z(s,t) = S M GHz MT TT
P1 x (t)  TMG1x
g11 
 
g12 
G1 x   
g13 


g
 14 
GH x
 g11
g
  21
 g 31

 g 41
g12
g 22
g13
g 23
g 32
g 42
g 33
g 43
g14 
g 24 
g 34 

g 44 
Hermite Surface Matrix
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Upper left = x-coordinates of surface
Upper right = x derivatives in t at corners
Lower left = x derivatives in s at corners
Lower right = twist at corners
Rendering surfaces
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Can use iterative methods in s and t
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Solve surface at points Q(s, t) and connect points
with quadrilaterals
Expensive because iterating for small s and t
results in many cubic surface evaluations
Forward Differencing
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Our old friend…
Because we can differentiate cubic curves three
times, we can increment all derivatives
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x += Dx
D x += D2x
D2x += D3x
Surface Rendering
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Subdivision
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As with cubic curves, Bezier cubic surface easily
supports subdivision
Subdivision ceases when plane described by one
quarter of the surface is nearly coplanar with the
other three-fourths
Watch for abutting quadrilaterals that don’t match
up
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This happens when different levels of subdivision are
applied to adjoining patches