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

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

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

4. Cindy Grimm Previous work Subdivision surfaces –Constructive method (arbitrary topology) –Induces local parameterization –C 1 continuity, higher order harder Patches –“Stitch” together n-sided patches Requires constraints on control points

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

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

7. Cindy Grimm 4n-sided polygon Vertices of polygon become one point on surface –Ordering of edges not same as ordering on polygon 2 0 1 3 6 5 7 4 0 1 65 4 3 2 7

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

9. Cindy Grimm Hyperbolic polygon Putting the two together: –Build 4n-sided polygon in hyperbolic disk Angles of corners sum to 2  h r    

10. Cindy Grimm 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

11. Cindy Grimm 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

12. Cindy Grimm 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

13. Cindy Grimm Different approach Cover the hyperbolic polygon with a manifold –Locally planar parameterization –Transition functions and blends between parameterizations 0 2 1 4 3 7 56

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

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

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

17. Cindy Grimm Manifold definition Constructive definition –Finite set A of non-empty subsets of R 2. Each subset c i is called a chart. –A set of subsets U ii =c i Empty, union of disjoint subsets. –Transition functions between subsets Reflexive Symmetric Transitive s

18. Cindy Grimm 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

19. Cindy Grimm 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

20. Cindy Grimm Choice of charts 2N+2 –One interior (unit disk) –One for each edge (unit square) –One “vertex” (unit disk) 0 2 1 4 3 7 56

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

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

23. Cindy Grimm 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

24. Cindy Grimm 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

25. Cindy Grimm Examples

26. Cindy Grimm 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 –C k for any k

27. Cindy Grimm Tessellation Edge Inside Vertex

28. Cindy Grimm User interface Click and drag

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