1. 1 Spherical manifolds for hierarchical surface modeling Cindy Grimm

2. 2 Graphite 2005, 12/1/2005 Goal Organic, free-form Adaptive Can add arms anywhere Hierarchical Move arm, hand moves C k analytic surface Build in pieces

3. 3 Graphite 2005, 12/1/2005 Overview Building an atlas for a spherical manifold Embedding the atlas Surface modeling Sketch mesh Adaptive editing

4. 4 Graphite 2005, 12/1/2005 Traditional atlas definition Given: Sphere (manifold) M Construct: Atlas A Chart Region U c in M (open disk) Region c in R 2 (open disk) Function  c taking U c to c Inverse Atlas is collection of charts Every point in M in at least one chart Overlap regions M

5. 5 Graphite 2005, 12/1/2005 Operations on the sphere How to represent points, lines and triangles on a sphere? Point is (x,y,z) Given two points p, q, what is (1-t) p + t q? Solution: Gnomonic projection Project back onto sphere Valid in ½ hemisphere Line segments (arcs) Barycentric coordinates in spherical triangles Interpolate in triangle, project All points such that… p q (1-t)p + tq

6. 6 Graphite 2005, 12/1/2005 Chart on a sphere Chart specification: Center and radius on sphere U c Range c = unit disk Simplest form for  c Project from sphere to plane (stereographic) Adjust with projective map Affine 1 c -1 Uc

7. 7 Graphite 2005, 12/1/2005 Defining an atlas Define overlaps computationally Point in chart: evaluate  c Coverage on sphere (U c domain of chart) Define in reverse as  c -1 =M D -1 (M W -1 (D)) D becomes ellipse after warp, ellipsoidal on sphere Can bound with cone normal

8. 8 Graphite 2005, 12/1/2005 Embedding the manifold Write embed function per chart (polynomial) Write blend function per chart (B-spline basis function) k derivatives must go to zero by boundary of chart Guaranteeing continuity Normalize to get partition of unity Normalized blend function Proto blend function

9. 9 Graphite 2005, 12/1/2005 Final embedding function Embedding is weighted sum of chart embeddings Generalization of splines Given point p on sphere Map p into each chart Blend function is zero if chart does not cover p Continuity is minimum continuity of constituent parts Embed Blend Map each chart

10. 10 Graphite 2005, 12/1/2005 Surface editing User sketches shape (sketch mesh) Create charts Embed mesh on sphere One chart for each vertex, edge, and face Determine geometry for each chart (locally)

11. 11 Graphite 2005, 12/1/2005 Charts Optimization Cover corresponding element on sphere Don’t extend over non-neighboring elements Projection center: center of element Map neighboring elements via projection Solve for affine map Face: big as possible, inside polygon Use square domain, projective transform for 4-sided Face Face charts

12. 12 Graphite 2005, 12/1/2005 Edge and vertex Edge: cover edge, extend to mid-point of adjacent faces Vertex: Cover adjacent edge mid points, face centers Edge Vertex Edge charts Vertex charts

13. 13 Graphite 2005, 12/1/2005 Defining geometry Fit to original mesh (?) 1-1 correspondence between surface and sphere Run subdivision on sketch mesh embedded on sphere (no geometry smoothing) Fit each chart embedding to subdivision surface Least-squares Ax = b

14. 14 Graphite 2005, 12/1/2005 Summary C K analytic surface approximating subdivision surface Real time editing Works for other closed topologies Parameterization using manifolds, Cindy Grimm, International Journal of Shape modeling 2004

15. 15 Graphite 2005, 12/1/2005 Hierarchical editing Override surface in an area Add arms, legs User draws on surface Smooth blend No geometry constraints

16. 16 Graphite 2005, 12/1/2005 Adding more charts User draws new subdivision mesh on surface Only in edit area Simultaneously specifies region on sphere Add charts as before Problem: need to mask out old surface

17. 17 Graphite 2005, 12/1/2005 Masking function Alter blend functions of current surface Zero inside of patch region Alter blend functions of new chart functions Zero outside of blend area Define mask function  on sphere, Set to one in blend region, zero outside 1 0 1 0

18. 18 Graphite 2005, 12/1/2005 Defining mask function Map region of interest to plane Same as chart mapping Define polygon P from user sketch in chart Define falloff function f(d) -> [0,1] d is min distance to polygon Implicit surface Note: Can do disjoint regions Mask blend functions 0 1 d

19. 19 Graphite 2005, 12/1/2005 Patches all the way down Can define mask functions at multiple levels Charts at level i are masked by all j>i mask functions Charts at level i zeroed outside of mask region

20. 20 Graphite 2005, 12/1/2005 Summary Flexible modeling paradigm No knot lines, geometry constraints Not limited to subdivision surfaces Alternative editing techniques?