Visualizing High-Order
Surface Geometry
Pushkar Joshi, Carlo Séquin
U.C. Berkeley
Clarification
• This talk is NOT about a new CAD tool;
 but it describes a “Meta-CAD tool.”
• This talk is NOT about designing surfaces;
 it is about understanding smooth surfaces.
This Presentation
• Convey geometrical insights via a
visualization tool for basic surface patches.
• Give a thorough understanding of what effects
higher-order terms can produce in smooth surfaces.
Visualizing Shape at Surface Point
Shape of small patch centered at surface point
• Build intuition behind abstract geometric concepts
• Applications: differential geometry, smoothness
metrics, identifying feature curves on surfaces
Minimizing Curvature Variation
for Aesthetic Design
Pushkar Joshi, Ph.D. thesis, Oct. 2008
Advisor: Prof. Carlo Séquin
U.C. Berkeley
http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-129.html
Minimum Curvature Variation Curves, Networks, and
Surfaces for Fair Free-Form Shape Design
Henry P. Moreton, Ph.D. thesis, March 1993
Advisor: Prof. Carlo Séquin
U.C. Berkeley
http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/5219.html
Surface Optimization Functionals
MES
optimal shape
MVS
optimal shape
Minimize:
total curvature
 (
2
1
  2 )dA
2
MVScross
optimal shape
Minimize: variation of curvature

22
22
d11  dk
d2 2   dk1   dk 2 
 dk

 
  dA

  
 dA
de11  de
de2 2   de 2   de1 
 de
2
new terms
2
Open Questions
• What is the right way to measure “total curvature variation” ?
– Should one average in-line normal curvature in all directions ?
• How many independent 3rd degree terms are there ?
– Does MVScross capture all of them, – with the best weighting ?
– Gravesen et al. list 18 different 3rd-degree surface invariants !
• How do these functionals influence surface shapes ?
– Which functional leads to the fairest, most pleasing shape ?
• Which is best basis for capturing all desired effects ?
– What is the geometrically simplest way to present that basis ?
Draw inspiration from principal curvatures and directions,
which succinctly describe second-degree behavior.
Visualizing 2nd Degree Shape
Flat
Hyperbolic
Parabolic
Elliptic
Principal curvatures (κ1, κ2) and principal directions (e1, e2)
completely characterize second-order shape.
Can we find similar parameters for higher-order shape?
Understanding the 2nd Degree Terms
• Analyze surface curvature in a cylindrical coordinate system
centered around the normal vector at the point of interest.
• Observe: offset sine-wave behavior of curvature around you,
with 2 maxima and 2 minima in the principal directions.
z=n
Curvature as a function of
rotation angle around z-axis:
phase-shifted sine-wave: F2
plus a constant offset: F0
Polynomial Surface Patch
z(u,v) =
C0u3 + C1u2v + C2uv2 + C3v3
4 Parameters!
+
Q0u2 + Q1uv + Q2v2
+
L0u + L1v
+
(const.)
Ignore (for now)
Fourier Analysis of Height Field
zc(r,θ) = r3
[ C0cos3(θ)+ C1cos2(θ)sin(θ)+ C2cos(θ)sin2(θ)+ C3sin3(θ)]
zc(r,θ) = r3 [ F1 cos( θ + α ) + F3 cos(3( θ + α + β )) ]
2
2
2
1
1
1
1
2
3
4
5
6
=
1
2
3
4
5
6
+
1
1
1
1
2
2
2
zc(θ)
F1cos(θ+α)
2
3
4
5
F3cos(3(θ+α+β))
6
3rd Degree Shape Basis Components
F1 (amplitude)
α (phase shift)
F3 (amplitude)
β (phase shift)
Visualizing 3rd Degree Shape in Fourier Basis
F1 component
x2
A cubic surface
(2 F1 + 2 F3 )/2
=
+
F3 component
=
x2
Directions Relevant to 3rd Degree Shape
z
Maximum F1
component
Maximum F3
component
( 3 equally
spaced
directions)
GUI of the Visualization Tool
Polynomial
Coefficients
Surface near point of analysis
Fourier
Coefficients
Surface is modified by changing polynomial coefficients or Fourier coefficients.
Changing one set of coefficients automatically changes the other set.
Polynomial & Fourier Coefficients
z(r, θ) = r3 [ J cos3θ + I sin3θ + H cos2θ sinθ + G cosθ sin2θ ]
+
r2 [ F cos2θ + E sin2θ + D cosθ sinθ ]
+
r [ C cosθ + B sinθ ]
+
A
(equivalent)
z(r, θ) = r3 [ F3_1 cos(θ + α) + F3_3 cos3(θ + α + β) ]
+
r2 [ F2_0 + F2_2 cos2(θ + γ) ]
+
r [ F1_1 cos(θ + δ) ]
+
For the math see:
F0_0
Joshi’s PhD thesis
3rd Degree Shape Edits (Sample Sequence)
(a)
(b)
(c)
(d)
(e)
(f)
Visualizing the Properties of a Surface Patch
Quadratic overlaid on cubic
Visualizing the Properties of a Surface Patch
Arrows indicate significant directions
Visualizing the Properties of a Surface Patch
Inline curvature derivative plot
Visualizing the Properties of a Surface Patch
Cross curvature derivative plot
3rd Degree Shape Parameters for General
Surface Patch
κ (θ)
n
κn(θ)
In-line curvature derivative
 n '( ) 
(Mehlum-Tarrou 1998)
 F1 cos(   )  F3 cos(3(     ))
Recap: 3rd Degree Shape Parameters
2nd Degree: κ1, κ2, φ
F0 = (κ1+κ2)/2
F2 = (κ1–κ2)/2
3rd Degree: F1, α, F3, β
The F1 and F3 components
relate to curvature derivatives.
Higher-Order Shape Bases
4th degree: F0
5th degree: F1
F2
F3
F4
F5
Application
Summary
Visualize 3rd degree basis shapes (using polynomial height field)
Develop theory of high-order basis shapes (Fourier coefficients)
Visualize higher-order (4th degree and 5th degree) basis shapes