1. Shape Space Exploration of Constrained Meshes Yongliang Yang, Yijun Yang, Helmut Pottmann, Niloy J. Mitra

2. Shape Space Exploration of Constrained Meshes Meshes and Constraints Meshes as discrete geometry representations Constrained meshes for various applications

3. Yas Island Marina Hotel Abu Dhabi Architect: Asymptote Architecture Steel/glass construction: Waagner Biro

4. Shape Space Exploration of Constrained Meshes Constrained Mesh Example (1) Planar quad (PQ) meshes [Liu et al. 2006]

5. Shape Space Exploration of Constrained Meshes Constrained Mesh Example (2) Circular/conical meshes [Liu et al. 2006]

6. Shape Space Exploration of Constrained Meshes

7. Problem Statement Given: single input mesh with a set of non-linear constraints in terms of mesh vertices Goal:  explore neighboring meshes respecting the prescribed constraints  based on different application requirements, navigate only the desirable meshes according to given quality measures

8. Shape Space Exploration of Constrained Meshes Example input meshes found via exploration

9. Shape Space Exploration of Constrained Meshes Basic Idea Exploration of a high dimensional manifold  Meshes with same connectivity are mapped to points  Constrained meshes are mapped to points in a manifold M  Extract and explore the desirable parts of the manifold M

10. Shape Space Exploration of Constrained Meshes Map Mesh to Point The family of meshes with same combinatorics Mesh point Deformation field d applied to the current mesh x yields a new mesh x + d Distance measure

11. Shape Space Exploration of Constrained Meshes Constrained mesh manifold M:  represents all meshes satisfying the given constraints Individual constraint  defines a hypersurface in Constrained Mesh Manifold

12. Shape Space Exploration of Constrained Meshes Constrained Mesh Manifold Involving m constraints in M is the intersection of m hypersurfaces  dimension D-m (tangent space)  codimension m (normal space)

13. Shape Space Exploration of Constrained Meshes PQ mesh manifold M: Constraints (planarity per face)  each face (signed diagonal distance)  deviation from planarity  10mm allowance for 2m x 2m panels Example: PQ Mesh Manifold represents all PQ meshes

14. Shape Space Exploration of Constrained Meshes Tangent Space starting mesh Geometrically, intersection of the tangent hyperplanes of the constraint hypersurfaces

15. Shape Space Exploration of Constrained Meshes Walking on the Tangent Space

16. Shape Space Exploration of Constrained Meshes Better Approximation ? Better approximation - 2 nd order approximant curved path consider the curvature of the manifold

17. Shape Space Exploration of Constrained Meshes a simple idea m hypersurfaces: E i = 0 (i=1, 2,..., m) osculating paraboloid S i the intersection of all osculating paraboloids:  hard to compute  not easy to use for exploration

18. Shape Space Exploration of Constrained Meshes Compute Osculant Generalization of the osculating paraboloid of a hypersurface: osculant Has the following form: Second order contact with each of the constraint hypersurfaces

19. Shape Space Exploration of Constrained Meshes 2 nd order contact amounts to solving linear systems

20. Shape Space Exploration of Constrained Meshes Walking on the Osculant

21. Shape Space Exploration of Constrained Meshes Mesh Quality? Osculant respects only the constraints Quality measures based on application  Mesh fairness: important for applications like architecture Extract the useful part of the manifold

22. Shape Space Exploration of Constrained Meshes Extract the Good Regions Abstract aesthetics and other properties via functions F(x) defined on Restricting F(x) to the osculant S(u) yields an intrinsic Hessian of the function F

23. Shape Space Exploration of Constrained Meshes Commonly used Energies Fairness energies  smoothness of the poly-lines Orthogonality energy  generate large visible shape changes

24. Shape Space Exploration of Constrained Meshes Applications

25. Shape Space Exploration of Constrained Meshes Spectral Analysis Good (desirable) subspaces to explore 2D-slice of design space

26. Shape Space Exploration of Constrained Meshes 2D Subspace Exploration

27. Shape Space Exploration of Constrained Meshes Handle Driven Exploration

28. Shape Space Exploration of Constrained Meshes stiffness analysis

29. Shape Space Exploration of Constrained Meshes Circular Mesh Manifolds Circular Meshes (discrete principal curvature param.)  Each face has a circumcircle

30. Shape Space Exploration of Constrained Meshes moving out into space

31. Shape Space Exploration of Constrained Meshes

32. Combined Constraint Manifolds

33. Shape Space Exploration of Constrained Meshes Future Work multi-resolution framework osculant surfaces  update instead of recompute (quasi-Newton) other ways of exploration interesting curves and 2-surfaces in M, …. applications where handle-driven deformation doesn’t really work (because of low degrees of freedom): form-finding

34. Shape Space Exploration of Constrained Meshes Acknowledgements Bailin Deng, Michael Eigensatz, Mathias Höbinger, Alexander Schiftner, Heinz Schmiedhofer, Johannes Wallner Funding agencies: FWF, FFG Asymptote Architecture

35. Shape Space Exploration of Constrained Meshes What Do We Gain?