Discrete Willmore Flow Alexander Bobenko and Peter Schrder Goals of - PowerPoint PPT Presentation

Discrete Willmore Flow Alexander Bobenko and Peter Schröder

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 � s � = � � H 4 � � k 1 � k 2 � 2 dA � � H: mean curvature=(k 1 +k 2 )/2 � K: Gaussian curvature= k 1 k 2 � k 1 , k 2 : are principle curvatures

Willmore Energy 2 � K � dA = 1 E w � s � = � � H 4 � � k 1 � k 2 � 2 dA � � Conformally invariant (M ö bius transforms) – Mostly just scale invariance that counts � Minimizing energy minimizes k 1 2 +k 2 2 � Special case of physics for thin structures � Fourth order flow—allows G 1 boundary conditions

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

Discretizations of Energy � Yoshizawa and Belyaev – Integrand not always positive 2 � K � dA = 1 E w � s � = � � H 4 � � k 1 � k 2 � 2 dA

Discrete formulation � Formulated for 2-manifolds with boundary i � 2 � i � vertices � � � j � e ij � M ö bius invariant � Positive � Zero if points lie on a sphere

Why is this Willmore flow? � Let D � be a triangulated neighborhood of a regular vertex W � D � � lim W s � D � � = R �� 0 � R is independent of curvatures � R � 1 � R=1 if two edges align with curvature lines

Details � Nice formulations of derivatives to reduce system size � Can handle singularities due to cocircular points – Uses M ö bius invariance

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 )

Solving � Forward Euler – Small timesteps � Backward Euler – Nonlinear system � In practice – Linearize gradient � 1 � t I � L � t ��� x � t � = �� E � t � – L is nxn with an entry per row per neighbor

Icosahedron � 24 steps

Cat head (fixed boundary)

Cat head (free boundary)

Torus

Pipe

Hole filling

Hole filling

Smoothing

Open questions � Error analysis and convergence analysis � Convergence to lines of curvature – Extend to quads