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

2 2

1 2( )dA

Minimize: total curvature

MES optimal shape

2 2

1 2

1 2

d ddA

de de

MVS optimal shape

Minimize: variation of curvature

MVScross

optimal shape

dAde

dk

de

dk

de

dk

de

dk2

1

2

2

2

1

2

2

2

2

1

1

new termsnew 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 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 Parabolic

Hyperbolic 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.

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) =

C0u3 + C1u2v + C2uv2 + C3v3

+

Q0u2 + Q1uv + Q2v2

+

L0u + L1v

+

(const.)

4 Parameters!

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 )) ]

1 2 3 4 5 6

2

1

1

2

1 2 3 4 5 6

2

1

1

2

1 2 3 4 5 6

2

1

1

2

= +

F1cos(θ+α) F3cos(3(θ+α+β))zc(θ)

3rd Degree Shape Basis Components

F3 (amplitude) β (phase shift)

F1 (amplitude) α (phase shift)

Visualizing 3rd Degree Shape in Fourier Basis

= + =

x2

x2

A cubic surface

F1 component

F3 component

(2 F1 + 2 F3 )/2

Directions Relevant to 3rd Degree Shape

Maximum F1 component

Maximum F3 component ( 3 equally spaced directions)

z

GUI of the Visualization Tool

PolynomialCoefficients

Fourier CoefficientsSurface 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 Coefficientsz(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

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) ] +

F0_0

(equivalent)

For the math see: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

(Mehlum-Tarrou 1998)

3rd Degree Shape Parameters for General Surface Patch

In-line curvature derivative

'( )n

κn(θ) κn(θ)

1 3cos( ) cos(3( ))F F

Recap: 3rd Degree Shape Parameters

2nd Degree: κ1, κ2, φF0 = (κ1+κ2)/2 F2 = (κ1–κ2)/2

3rd Degree: F1, α, F3, βThe F1 and F3 componentsrelate to curvature derivatives.

Higher-Order Shape Bases

4th degree: F0

5th degree: F1

F2

F3 F5

F4

Application

SummaryVisualize 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