Visualizing High-Order Surface Geometry

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Visualizing High-Order Surface Geometry

• Pushkar Joshi, Carlo Séquin
• U.C. Berkeley

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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.

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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

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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/EE
  CS-2008-129.html

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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/1993/52
  19.html

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Surface Optimization Functionals
MES optimal shape
Minimize total curvature
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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.

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Visualizing 2nd Degree Shape
Flat
Parabolic
Hyperbolic
Elliptic
Principal curvatures (?1, ?2) and principal
directions (e1, e2) completely characterize
second-order shape.
Can we find similar parameters for higher-order
shape?
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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.

Curvature as a function of rotation angle around
z-axis
z n
phase-shifted sine-wave F2plus a constant
offset F0
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Polynomial Surface Patch

• z(u,v)
• C0u3 C1u2v C2uv2 C3v3
• Q0u2 Q1uv Q2v2
• L0u L1v
• (const.)

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Fourier Analysis of Height Field

• zc(r,?) r3 C0cos3(?) C1cos2(?)sin(?)
  C2cos(?)sin2(?) C3sin3(?)
• zc(r,?) r3 F1 cos( ? a ) F3 cos(3( ? a
  ß ))

F1cos(?a)
F3cos(3(?aß))
zc(?)
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3rd Degree Shape Basis Components
F3 (amplitude) ß (phase shift)
F1 (amplitude) a (phase shift)
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Visualizing 3rd Degree Shape in Fourier Basis
F1 component
x2
A cubic surface
(2 F1 2 F3 )/2



F3 component
x2
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Directions Relevant to 3rd Degree Shape
z
Maximum F1 component
Maximum F3 component ( 3 equally spaced
directions)
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GUI of the Visualization Tool
Fourier Coefficients
PolynomialCoefficients
Surface near point of analysis
Surface is modified by changing polynomial
coefficients or Fourier coefficients. Changing
one set of coefficients automatically changes the
other set.
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Polynomial Fourier Coefficients
z(r, ?) r3 J cos3? I sin3? H cos2? sin?
G cos? sin2? r2 F cos2? E sin2? D
cos? sin? r C cos? B sin? A
(equivalent)
z(r, ?) r3 F3_1 cos(? a) F3_3 cos3(? a
ß) r2 F2_0 F2_2 cos2(? ?) r
F1_1 cos(? d) F0_0
For the math see Joshis PhD thesis
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3rd Degree Shape Edits (Sample Sequence)
(a) (b)
(c)
(d) (e)
(f)
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Visualizing the Properties of a Surface Patch
Quadratic overlaid on cubic
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Visualizing the Properties of a Surface Patch
Arrows indicate significant directions
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Visualizing the Properties of a Surface Patch
Inline curvature derivative plot
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Visualizing the Properties of a Surface Patch
Cross curvature derivative plot
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3rd Degree Shape Parameters for General Surface
Patch

• In-line curvature derivative

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Recap 3rd Degree Shape Parameters
2nd Degree ?1, ?2, f F0 (?1?2)/2 F2
(?1?2)/2
3rd Degree F1, a, F3, ßThe F1 and F3
componentsrelate to curvature derivatives.
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Higher-Order Shape Bases
4th degree F0
F2
F4
5th degree F1
F3
F5
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Application
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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