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Discrete Willmore Flow
Alexander Bobenko and Peter Schröder
Goals of Paper
Present a technique for smoothing surfaces Advance idea of discrete geometry vs.
Discrete Willmore Flow Alexander Bobenko and Peter Schrder Goals of - - PowerPoint PPT Presentation
Discrete Willmore Flow Alexander Bobenko and Peter Schrder Goals of - - PowerPoint PPT Presentation
Discrete Willmore Flow Alexander Bobenko and Peter Schrder Goals of Paper Present a technique for smoothing surfaces Advance idea of discrete geometry vs. discretizations of continuous geometry Willmore Energy 2 K dA = 1 E w
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Willmore Energy
- H: mean curvature=(k1+k2)/2
K: Gaussian curvature= k1k2 k1, k2: are principle curvatures
E ws=H
2KdA= 1
4k1k2
2dA
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Willmore Energy
- Conformally invariant (Möbius transforms)
– Mostly just scale invariance that counts
Minimizing energy minimizes k1
2+k2 2
Special case of physics for thin structures Fourth order flow—allows G1 boundary conditions
E ws=H
2KdA= 1
4k1k2
2dA
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Discrete Geometry
Traditional approach
– Finite elements or finite differences – Lose many of the guarantees of the continuous – Finite elements are well understood
Alternative
– Formulate discrete analogs – Respect symmetries/invariants
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Discretizations of Energy
Yoshizawa and Belyaev
– Integrand not always positive E ws=H
2K dA= 1
4k1k2
2dA
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Discrete formulation
Formulated for 2-manifolds with boundary
- Möbius invariant
Positive Zero if points lie on a sphere
- ivertices
eij
j
i 2
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Why is this Willmore flow?
Let D
be a triangulated neighborhood of a
regular vertex
R is independent of curvatures R1 R=1 if two edges align with curvature lines
lim W D W sD=R
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Details
Nice formulations of derivatives to reduce system
size
Can handle singularities due to cocircular points
– Uses Möbius invariance
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Boundary Conditions
G1
– Fixed positions and normals – Not necessarily on boundaries
Free
– Add a point at infinity per boundary – Extra terms in energy:
- angle of boundary triangle (2)
Angle between boundary edges (3)
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Solving
Forward Euler
– Small timesteps
Backward Euler
– Nonlinear system
In practice
– Linearize gradient – L is nxn with an entry per row per neighbor 1 t ILt xt=Et
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Icosahedron
24 steps
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Cat head (fixed boundary)
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Cat head (free boundary)
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Torus
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Pipe
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Hole filling
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Hole filling
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Smoothing
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Open questions
Error analysis and convergence analysis Convergence to lines of curvature
– Extend to quads