Summer Seminar

1. Summer Seminar Lubin Fan 2011-07-07

2. Discrete Differential Geometry • Circular arc structures • Discrete Laplacians on General Polygonal Meshes • HOT: Hodge-Optimized Triangulations • Spin Transformations of Discrete Surfaces Example-Based Simulation • Frame-based Elastic Models (TOG) • Sparse Meshless Models of Complex Deformable Objects • Example-Based Elastic Materials

3. Circular Arc Structures Pengbo Bo1,2 Helmut Pottmann2,3 Martin Kilian2 Wenping Wang1 Johannes Wallner2,4 1Univ. Hong Kong 2TU Wien 3KAUST 4TU Graz

4. Authors Helmut Pottmann KAUST Vienna University of Technology Pengbo Bo Postdoctoral Fellow Univ. Hong Kong Martin KilianRA Vienna University of Technology Wenping Wang Professor Univ. Hong Kong Johannes Wallner Professor Graz University of Technology Vienna University of Technology

5. Architectural Geometry • The most important guiding principle for freeform architecture • Balance • Cost efficiency • Adherence to the design intent • Key issue • Simplicity of supporting and connecting elements as well as repetition of costly parts Node complexity

6. Previous Work • Nodes optimization • [Liu et al. 2006; Pottmann et al. 2007] for quad meshes • [Schiftner et al. 2009] for hexagonal meshes • Rationalization with single-curved panel • [Pottmann et al. 2008] • Repetitive elements • [Eigensatz et al. 2010] • [Singh and Schaefer 2010] and [Fu et al. 2010] • The aesthetic quality is reduced if the number of repetitions increases.

7. This Work • Propose the class of Circular Arc Structures (CAS) • Properties • Smooth appearance, congruent nodes, and the simplest possible elements for the curved edges • Do not interfere with an optimized skin panelization. • Contributions • freeform surfaces may be rationalized using CAS • repetitions not only in nodes, but also in radii of circular edges • extend to fully three-dimensional structures • have nice relations to discrete differential geometry and to the sphere geometries

8. Circular Arc Structures • Definition A circular arc structure consists of 2D mesh combinatorics (V, E), where edges are realized as circular arcs, such that in each vertex the adjacent arcs touch a common tangent plane. We require congruence of interior vertices, and we consider the following three cases: • Hexagonal CAS have valence 3 vertices. Angles between edges equal 120 degrees; • Quadrilateral CAS have valence 4 vertices. Angles between edges have values α, π − α, α, π − α, if one walks around a vertex; • Triangular CAS have valence 6 vertices. Angles between edges equal 60 degrees.

9. Circular Arc Structures • Data Structure • Target Functional • Deviation • Smoothness • Geometric consistency • Regularization • Angles

10. Circular Arc Structures • Generalizations • Singularities • Supporting Elements • Add condition

11. CAS with Repetitive Elements • Radius Repetitive • Definition A quadrilateral CAS is radius-repetitive along a flow line, if the radius of its edges is constant. It is transversely radius-repetitive for a pair of neighboring ‘parallel’ flow lines, if the edges which connect these flow lines have constant radius. • Condition

12. Cyclidic Structure • Cyclidic CAS • Offsets • Offsetting operation of cyclidic CAS is well defined

13. Results

14. Conclusions • Limitations • Loss of shape flexibility when additional geometric conditions are imposed. • The introduction of T-junctions • This Work • Shown the applicability of CAS • Demonstrated special CAS have more properties which are relevant for freeform building construction • Future Work • Explore more application

15. Discrete Laplacians on General Polygonal Meshes Marc Alexa1 Max Wardetzky2 1TU Berlin 2Universitaat Gottingen

16. Authors Marc Alexa Professor Electrical Engineering and Computer Science TU Berlin Max Wardetzky Assistant Professor Heading the Discrete Differential Geometry Lab Universitaat Gottingen

17. This Work • Discrete Laplacianon surface with arbitrary polygonal faces • Non-planar & non-convex polygons • Mimic structural properties of the smooth Laplace-Beltrami operator • Motivation • Non-triangular polygons are widely used in geometry processing

18. Related Work • Geometric discrete Laplacians • Cotan formula [Pinkall and Polthier 1993] • The last decade has brought forward several parallel developments… • Application • Mesh parameterization • Fairing • Denoising • Manipulation • Compression • Shape analysis • …

19. Discrete Laplacian Framework • Setup • An oriented 2-manifold mesh M, possibly with boundary, with vertex set V , edge set E, and face set F . We allow for faces that are simple, but possibly non-planar, polygons in R3. • Work with oriented halp-edge • EI, inner edges; EB boundary edge • Algebraic approach to discrete Laplacian • M0 • M1

20. Desiderate • Locality • Maintain locality by only working with diagonal matrices M0 and by requiring that M1 is defined per face in the sense that • Symmetry : L = LT • Positive semi-definiteness • M0 & Mf are positive definiteness. • Linear precision • Scale invariance • Convergence

21. Vector Area & Maximal Projection • Vector Area • Maximal Projection • Mean Curvature Maximal Projcetion

22. A family of discrete Laplacians • [Perot and Suvramanian 2007] —— pre-Laplacians —— positive semi-definite

23. Implementation • Construct 3 matrices • Diagonal matrx, M0 • Coboundary matrix, d • dep = ±1 if e = ±eqp and dep = 0 • M1 • Assembled per face: Mf

24. Results & Application • Implicit mean curvature flow • Parameterization

25. Results & Application • A planarizing flow

26. Results & Application • Thin plate bending

27. Conclusion • This Work • presents here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non-planar and non-convex polygons. • Feature Work • How to replace this combinatorial term by a more geometric one

28. Spin Transformation of Discrete Surface Keenan Crane1 Ulrich Pinkall2 Peter Schroder1 1California Institute of Technology 2TU Berlin http://users.cms.caltech.edu/~keenan/project_spinxform.html

29. Authors Ulrich Pinkall Geometry Group Institute of mathematics TU Berlin Keenan Crane PhD Student California Institute of Technology Peter SchroderProfessor Director of the Multi-Res Modeling Group California Institute of Technology

30. This Work • Spin Transformation • A new method for computing conformal transformations of triangle meshes in R3 • Consider maps into the quaternions H

31. Related Work • Deformation • Local coordinate frame [Lipman et al. 2005, Paries et al. 2007] • Cage-based editing [Lipman et al. 2008] • Surface parametrization • Prescribe values at vertices that directly control the rescaling of the metric[Ben-Chen et al. 2008; Yang et al. 2008; Springborn et al. 2008].

32. Quaternion • Definition • The quaternions H can be viewed as a 4D real vector space with basis {1, i, j, k} along with the non-commutative Hamilton product, which satisfies the relationships i2 = j2 = k2 = ijk = −1. • The imaginary quaternions Im H are elements of the 3D subspace spanned by {i, j, k}. • q = a + bi + cj + dk, q = a - bi - cj – dk • Rotation of a vector , , (Similarity Transformation) • Calculus • Map f : M -> ImH • Differential df : TM -> ImH

33. Spin Transformations

34. Spin Transformations • Integrable Condition [Kamberov et al. 1998] • D , Quaternionic Dirac Operator • Eigenvalue Problem

35. Spin Transformations • Procedure • Pick a scalar function ρ on M • Solve an eigenvalue problem for the similarity transformation λ • Sovle a linear system for the new surface

36. Discretization • Discrete Dirac Operator

37. Discretization • Scalar Multiplication • Discretized Spin Transformations

38. Application • Painting Curvature

39. Application • Arbitrary Deformation

40. Conclusion • This Work • Our discretization of the integrability condition (D − ρ)λ = 0 provides a principled, efficient way to construct conformal deformations of triangle meshes in R3. • Future Work • D is expressed in terms of extrinsic geometry it can be used to compute normal information, mean curvature, and the shape operator.

41. California Institute of Technology HOT: Hodge-Optimized Triangulations Patrick Mullen Pooran Memari Fernando de Goes Mathieu Desbrun

42. This Work • “Good” dual • Motivation • Fluid simulation • This work • Hodge-optimized triangulation

43. Previous Work • Delaunay / Voronoi pairs • [Meyer et al. 2003] • [Perot and Subramanian 2007] • [Elcott et al. 2007] • Drawbacks • Circumcenter lies outside its associated tetrahedron • Inability to choose the position of dual mesh • Too restrictive in many practical situations

44. Results

45. Results

46. 1University of British Columbia, Vancouver, CANADA 2University of Grenoble 3INRIA 4LJK – CNRS Frame-based Elastic Models Benjamin Gilles1 Guillaume Bousquet2,3,4 Francois Faure2,3,4 Dinesh K. Pai1

47. Authors Guillaume Bousquet Second year PhD student University of Grenoble Laboratoire Jean KuntzmannINRIA Benjamin Gilles Post-doctoral Fellow Sensorimotor Systems Lab Department of Computer ScienceUniversity of British Columbia François Faure Assistant Professor University of Grenoble Laboratoire Jean KuntzmannINRIA Dinesh K. Pai Professor Sensorimotor Systems Lab Department of Computer ScienceUniversity of British Columbia

48. Deformable Models [Terzopoulos et al. 1988] • Application • Computer animation • Animating characters, Soft objects, … • Approaches • Physically based deformation • Skinning

49. Physically based deformation [Nealen et al. 2005] • Finite Element Method • Lagrangian models of deformable objects • Two main method • Mesh-based methods • Meshless methods • Pros • Physical realism • Cons • Expensive • Difficult to use

50. Physically based deformation • Lagrangian mechanics • Simulation loop