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 Advisor: Prof. Carlo Séquin U.C. Berkeley

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

Surface Optimization Functionals Minimize: total curvature MES optimal shape MVS optimal shape Minimize: variation of curvature MVS cross optimal shape new terms

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 3 rd degree terms are there ? – Does MVS cross capture all of them, – with the best weighting ? – Gravesen et al. list 18 different 3 rd -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 2 nd Degree Shape FlatParabolic Hyperbolic Elliptic Principal curvatures (κ 1, κ 2 ) and principal directions (e 1, e 2 ) completely characterize second-order shape. Can we find similar parameters for higher-order shape?

Understanding the 2 nd 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: F2 plus a constant offset: F0

Polynomial Surface Patch z(u,v) = C 0 u 3 + C 1 u 2 v + C 2 uv 2 + C 3 v 3 + Q 0 u 2 + Q 1 uv + Q 2 v 2 + L 0 u + L 1 v + (const.) 4 Parameters! Ignore (for now)

Fourier Analysis of Height Field z c (r,θ) = r 3 [ C 0 cos 3 (θ)+ C 1 cos 2 (θ)sin(θ)+ C 2 cos(θ)sin 2 (θ)+ C 3 sin 3 (θ)] z c (r,θ) = r 3 [ F 1 cos( θ + α ) + F 3 cos(3( θ + α + β )) ] = + F 1 cos(θ+α)F 3 cos(3(θ+α+β)) zc(θ)zc(θ)

3 rd Degree Shape Basis Components F 3 (amplitude) β (phase shift) F 1 (amplitude) α (phase shift)

Visualizing 3 rd Degree Shape in Fourier Basis =+= x2x2 x2x2 A cubic surface F 1 component F 3 component (2 F F 3 )/2

Directions Relevant to 3 rd Degree Shape Maximum F 1 component Maximum F 3 component ( 3 equally spaced directions) z

GUI of the Visualization Tool Polynomial Coefficients Fourier Coefficients 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.

Polynomial & Fourier Coefficients z(r, θ) = r 3 [ J cos 3 θ + I sin 3 θ + H cos 2 θ sinθ + G cosθ sin 2 θ ] + r 2 [ F cos 2 θ + E sin 2 θ + D cosθ sinθ ] + r [ C cosθ + B sinθ ] + A z(r, θ) = r 3 [ F 3_1 cos(θ + α) + F 3_3 cos3(θ + α + β) ] + r 2 [ F 2_0 + F 2_2 cos2(θ + γ) ] + r [ F 1_1 cos(θ + δ) ] + F 0_0 (equivalent) For the math see: Joshi’s PhD thesis

3 rd 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

(Mehlum-Tarrou 1998) 3 rd Degree Shape Parameters for General Surface Patch In-line curvature derivative κn(θ)κn(θ) κn(θ)κn(θ)

Recap: 3 rd Degree Shape Parameters 2 nd Degree: κ 1, κ 2, φ F 0 = (κ 1 +κ 2 )/2 F 2 = (κ 1 –κ 2 )/2 3 rd Degree: F 1, α, F 3, β The F 1 and F 3 components relate to curvature derivatives.

Higher-Order Shape Bases 4 th degree: F0 5 th degree: F1 F2 F3F5 F4

Application

Summary Visualize 3 rd degree basis shapes (using polynomial height field) Develop theory of high-order basis shapes (Fourier coefficients) Visualize higher-order (4 th degree and 5 th degree) basis shapes