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Surfaces

Surfaces. Reporter: Gang Xu May 30, 2006. Surfaces Session. Aug 1, 2006, 8:30-10:15 am Session Chair: Leif.P Kobbelt. Papers. Real-Time GPU Rendering of Piecewise Algebraic Surfaces Charles Loop , Jim Blinn ( Microsoft Research ) Point-Sampled Cell Complexes

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Surfaces

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  1. Surfaces Reporter: Gang Xu May 30, 2006

  2. Surfaces Session • Aug 1, 2006, 8:30-10:15 am • Session Chair: Leif.P Kobbelt.

  3. Papers • Real-Time GPU Rendering of Piecewise Algebraic Surfaces • Charles Loop, Jim Blinn (Microsoft Research) • Point-Sampled Cell Complexes • Anders Adamson (Technische Universität Darmstadt), Marc Alexa (Technische Universität Berlin) • Geometric Modeling with Conical Meshes and Developable Surfaces • Yang Liu (University of Hong Kong), Helmut Pottmann, Johannes Wallner (Technische Universität Wien ), Wenping Wang (University of Hong Kong), Yong-Liang Yang (Tsinghua University) • Mesh Quilting For Geometric Texture Synthesis • Kun Zhou, Xin Huang, Xi Wang (Microsoft Research Asia), Yiying Tong, Mathieu Desbrun (California Institute of Technology), Baining Guo, Heung-Yeung Shum (Microsoft Research Asia)

  4. Paper One • Real-Time GPU Rendering of Piecewise Algebraic Surfaces • Direct rendering of Bézier tetrahedra by specially encoding coefficient data on vertices and rasterizing faces. A pixel shader then robustly solves the (up to fourth-order) equations using analytic techniques.

  5. Paper four • Mesh Quilting for Geometric Texture Synthesis • A novel geometry texture synthesis algorithm, called mesh quilting, that seamlessly synthesizes a 3D texture sample given in the form of a triangle mesh over arbitrary surfaces.

  6. Geometric modeling with conical meshes and developable surface Liu Yang, H.Pottman, J. Wallner, Wang Wen-Ping, Yang Yong-Liang

  7. Application Background(1) Requirement of architectural design with glass structures

  8. Application Background(2)

  9. Conical Meshes

  10. Previous Work (1) • Discrete differential geometry Course in Siggraph 2005

  11. Previous Work (2) • PQ meshes: quad meshes with planar faces, discrete counterpart of conjugate curve net works on surfaces (Sauer,1970)

  12. Previous work(3) Circular meshes: quad meshes that all quads possess a circumcircle, discrete counterpart of the network of principal curvature lines (Martin et al, 1986)

  13. Previous work(4) • Quad mesh • Computation of quad-dominant meshes from smoothed principal curvature lines (Alliez et al, 03) • Variational shape approximation (Cohen, 04)

  14. Previous work(5) • Developable surface • Arrangement of n planar quads in a single row.

  15. Previous work(6) • Surface for architecture and aesthetic design (Sequin, 2005)

  16. Contribution • Introduce conical meshes and demonstrate their superiority • PQ perturbation algorithm • How to produce conical meshes and circular meshes?

  17. PQ meshes(1) PQ perturbation: optimization problem with constraints

  18. PQ meshes(2) • A thin planar quad which converges to a straight line segment, above constraints serve to maintain convexity but they can’t express the planarity.

  19. PQ meshes(3) Two energy terms: Fair shape: Close to the original mesh:

  20. PQ mesh(4) • Objective function • Small or medium case: Sequential Quadratic Programming (SQP) • Large case: penalty method

  21. PQ mesh(5)

  22. Subdividing PQ meshes • Generate a PQ mesh from a coarse mesh : combine the PQ perturbation algorithm with a DS or CC algorithm

  23. Subdivision developable surfaces • Applying a curve subdivision rule to its boundaries • Subsequent application of PQ perturbation

  24. Conical Meshes(1) • Conical vertex: all the four planes meeting at the vertex are tangent to a common sphere (or cone of revolution)

  25. Conical Meshes(2) • Conical mesh: all of the its vertex of valance four are conical. • Conical mesh is the discrete counterpart of the system of principal curvature lines.

  26. Conical Meshes(3) • Types of conical mesh vertexes ellipticparabolichyperbolic

  27. Conical Meshes(4) • An angle criterion for conical meshes A vertex of a quad mesh is a conical vertex if and only if the angle balance is satisfied.

  28. Conical Meshes(5) • Offsetting conical meshes Offsetting all faces by a fixed distance leads again to a mesh with the same connectivity

  29. Conical Meshes(6) • The offsetting property is a characterizing property of conical meshes. • General Laguerre transformations map conical meshes to conical meshes • Mobius transformations map circular meshes to circular meshes

  30. Conical Meshes(7) • Neighboring axes(discrete surface normals) are co-planar, they are contained in a plane orthogonal to the mesh in a discrete sense.

  31. Conical Meshes(8) • An edge of the mesh, the discrete normals at its endpoints, and the corresponding edge of any offset mesh lie on a common plane.

  32. Conical Meshes(9) • Successive discrete normals of a conical mesh along a row or column are co-planar and therefore form a discrete developable surface.

  33. Conical Meshes(9) • If a subdivision process, which preserves the conical property, refines a conical mesh and in the limit produces a curve network on a surface, then this limit curve network is the network of principal curvature lines .

  34. Conical Meshes(10) • If there are meshes which are both circular and conical?

  35. Computing Conical/Circular Meshes • Similar with the PQ perturbation algorithm. • Similar with the PQ perturbation algorithm.

  36. Results(1)

  37. Results(2)

  38. Future work in the paper • Computation of special discrete surfaces in a principal mesh and especially conical mesh representation.

  39. Other future work(1) • There are many surfaces that their parameter curves are their lines of curvatures, such as surface of revolution, Enneper’s minimal surfaces.

  40. Other future work(2) • Change the constraint condition with angles to the constraint condition with edges. • 托勒密定理：設四邊形ABCD內接於圓，則有AB×CD + AD×BC=AB×BD。 註：在凸四邊形ABCD中，有AB×CD + AD×BC ≧AB×BD。等號成立的充要條件是ABCD為圓內接四邊形。

  41. Other future work(3) • Can we use the PQ perturbation algorithm and subdivision algorithm to modeling other special surfaces? • Architectural design using minimal surface

  42. Point-Sampled Cell Complexes Anders Adamson Marc Alexa TU Darmstadt TU Berlin

  43. Surface modeling method • NURBS (T-Spline): parameter domain • Subdivision surface: each face • PDE surface • FFD method • Physics-based method

  44. Contribution • Provide a versatile shape representation for modeling that allows prescribing features exactly

  45. Point-sampled surface • Defining a point-sampled surface is difficult. • It is usually defined as the level set of a smooth function.

  46. Definition of the Point surface(1)

  47. Definition of the Point surface(2) • Wendland’s radial functions Projection procedure:

  48. Definition of the Point surface(3) • The point set surface is defined as the set of stationary points under projection

  49. Cell complex(胞腔复形)(1) • 流形结构: N维拓扑形体只能由(N-1)维拓扑形体组成. • 非流形结构:N维拓扑形体不仅由有限个(N-1)维拓扑形体, 而且还包含(N-2),(N-3), …., 0维形体的信息.

  50. Cell complex(胞腔复形)(2)

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