Parameterizing N-holed Tori

1. Parameterizing N-holed Tori Cindy Grimm (Washington Univ. in St. Louis) John Hughes (Brown University) Cindy Grimm

2. Parameterizing n-holed tori • “Natural” method for parameterizing non-planar topologies • Constructive • Amenable to spline-like embedding • Control points • Local control • Polynomial Cindy Grimm

3. Outline • Related work • Patch approach • Topology • Related work • Hyperbolic approach • Manifold approach • Constructive approach to modeling topology • Embedding Cindy Grimm

4. Previous work • Subdivision surfaces • Constructive method (arbitrary topology) • Induces local parameterization • C1 continuity, higher order harder • Patches • “Stitch” together n-sided patches • Requires constraints on control points Cindy Grimm

5. 7 4 2 5 2 1 1 6 0 3 0 3 4 7 5 6 Topology • Building n-holed tori • Associate sides of 4n polygon Cindy Grimm

6. 7 4 2 5 1 6 0 3 2 1 3 0 4 7 5 6 4n-sided polygon • One loop through hole • a, a-1 • One loop around hole • b, b-1 • Repeat for n holes Cindy Grimm

7. 7 4 2 5 1 6 0 3 4n-sided polygon • Vertices of polygon become one point on surface • Ordering of edges not same as ordering on polygon 2 1 0 3 7 4 5 6 Cindy Grimm

8. Hyperbolic disk • Unit disk with hyperbolic geometry • Sum of triangle angles < 180 • Lines are circle arcs • Circles meet disk perpendicularly Cindy Grimm

9. b a w r h g Hyperbolic polygon • Putting the two together: • Build 4n-sided polygon in hyperbolic disk • Angles of corners sum to 2p Cindy Grimm

10. Associate edges • Associate edges • Tile disk with infinite copies • Example in 1D • Tile real line with (0,1] • Associate s with every point s+i • Result is a circle Cindy Grimm

11. Transition functions • Linear fractional transforms (LFTs) • Map disk to itself by “flipping” over an edge • Well-defined inverse • Combine • Scale, rotation, translation • Use many LFT to associate edges of polygon Cindy Grimm

12. Previous work • Hyperbolic geometry approach • A. Rockwood, H. Ferguson, and H. Park • J. Wallner and H. Pottmann • Define motion group • Define multi-periodic basis functions (cosine/sine) • Make edges match up Cindy Grimm

13. Different approach • Cover the hyperbolic polygon with a manifold • Locally planar parameterization • Transition functions and blends between parameterizations 2 1 0 7 3 4 5 6 Cindy Grimm

14. Different approach • Embed the manifold • Embedding function for each local parameterization • Splines, RBFs, etc. • Blend between local embeddings Cindy Grimm

15. Roadmap • Building a manifold • Constructive definition • Choice of charts, transition function • Embedding function • Local embedding functions • Blend functions • Tessellation • User interaction Cindy Grimm

16. s Manifold definition • Traditional: Locally Euclidean • Chart: Map from surface to plane • Induces overlap regions, transition functions Cindy Grimm

17. s Manifold definition • Constructive definition • Finite set A of non-empty subsets of R2. Each subset ci is called a chart. • A set of subsets • Uii=ci • Empty, union of disjoint subsets. • Transition functions between subsets • Reflexive • Symmetric • Transitive Cindy Grimm

18. Manifold definition • “Glue” points together using transition functions • A “point” on this manifold is a tuple of chart, 2D point pairs • If built from existing manifold, corresponds to point on existing manifold • Under certain technical assumptions, above definition (with points glued together using transition functions) is a manifold • No geometry Cindy Grimm

19. Hyperbolic polygon manifold • Use existing manifold (hyperbolic polygon with associated edges) to define charts, overlap regions, transitions • Constructed object will be a manifold • Many possible choices for charts • Minimal number • Unit square or unit disk Cindy Grimm

20. Choice of charts • 2N+2 • One interior (unit disk) • One for each edge (unit square) • One “vertex” (unit disk) 2 1 0 7 3 4 5 6 Cindy Grimm

21. Inside-edge Edge-inside Vertex-edge Edge-edge Edge-vertex Inside-vertex Vertex-inside Transition functions • Map from chart to polygon to chart • Check region, apply LFT Cindy Grimm

22. Status • Structure which is locally planar • Unit disk • Unit square • Equate points in each chart • Transition functions/overlap regions • Topology • No geometry Cindy Grimm

23. Embedding function • Define embedding function per chart • Any 2D->3D function, domain can be bigger than chart • Nice (but not necessary) if functions agree where they overlap • Define blend function per chart • Values, derivative zero by chart boundary • Radial or square B-spline basis function • Promote to function on manifold by setting equal to zero elsewhere Cindy Grimm

24. Embedding function • Divide by sum of chart blend functions to create a partition of unity • Ensure sum is non-zero • Continuity is minimum continuity of blend, embedding, and transition/chart functions Cindy Grimm

25. Examples Cindy Grimm

26. Remarks • Natural parameterization • Extract local planar parameterization • Spline-like embedding • Topology in manifold structure • Embedding structure independent of choice of planar embedding function • Local control • Rational polynomials • Ck for any k Cindy Grimm

27. Edge Vertex Inside Tessellation Cindy Grimm

28. User interface • Click and drag Cindy Grimm

29. Future work • Parameterize existing meshes, subdivision surfaces • Better embeddings • N-sided patches for inside, vertex charts • Alternative hyperbolic geometries • Klein-Beltrami Cindy Grimm