1 Adding charts anywhere Assume a cow is a sphere Cindy Grimm and John Hughes, “Parameterizing n-holed tori”, Mathematics of Surfaces X, 2003 Cindy Grimm, “Parameterization using Manifolds”, International Journal of Shape Modelling, 2004

2 Siggraph 2005, 8/1/ Motivation Surface modeling: Adding more degrees of freedom Don’t want to be restricted to existing charts/dof Computers support vectors (points in R n ) Functions on R n Splines, gradient descent algorithms Define functions directly on sphere? BRDFs, environment maps

3 Siggraph 2005, 8/1/ Observation We can define lots of meshes that are spheres Which one’s the “right” one? How do you compare two sphere meshes?

4 Siggraph 2005, 8/1/ Approach Find a computationally tractable way to represent some common domains E.g., sphere, torus, plane Doesn’t depend on any particular geometry Questions to answer: What is a point? (Convert from R n to domain) Movement in domain Barycentric coordinates, linear interpolation

5 Siggraph 2005, 8/1/ Once we have a domain… Add charts. Anywhere. Not restricted by existing atlas/construction Questions to answer: Chart specification? Chart function form?

6 Siggraph 2005, 8/1/ Applications Surface modeling, texture mapping Surface comparison, feature parameterization Sphere: environment maps, BRDFs Arbitrary chart placement Port existing tools (linear embedding functions, gradient descent algorithms, etc.) to any domain transparently Adaptive parameterization

7 Siggraph 2005, 8/1/ Outline Spherical domain Defining Adding charts Building embedding function (surface Torus, n-holed tori domains Torus: Tiled plane N-holed: Tiled plane, sort-of

8 Siggraph 2005, 8/1/ Atlas from mesh Embedded sketchVertex chartsSketch Surface Embed mesh in domain 1-1 correspondence Specifies both chart locations and chart embeddings

9 Siggraph 2005, 8/1/ In contrast… Approaches presented earlier created abstract manifold from scratch Specify transition functions/overlaps Chart functions derived Charts exactly match mesh connectivity This approach Define chart functions Transition functions/overlaps are derived Charts loosely tied to mesh connectivity Both approaches fit geometry to subdivision surface

10 Siggraph 2005, 8/1/ Representing a sphere Representing points x 2 +y 2 +z 2 -1 = 0 Average/blend points Project back onto sphere (Gnomonic projection) Valid in ½ hemisphere Barycentric coordinates in spherical triangles Interpolate in triangle, project Line segments (arcs)

11 Siggraph 2005, 8/1/ Charts Stereographic projection Sketch mesh charts Circles to circles Requires rotation to center at arbitrary point Both C  Images: mathworld.wolfram.com

12 Siggraph 2005, 8/1/ Charts, adjusting Raw projections M D don’t provide much control Map to consistent chart shape (e.g., unit disk) Add additional “warp” function M W Projective transform (scale, translate, rotate) Still C 

13 Siggraph 2005, 8/1/ Overlaps and transition functions Overlaps Chart inverse functions well-defined, C , over chart range/disk (but not over entire plane) Coverage of chart determined by inverse map Disk->sphere Transition functions Defined by composition  (  -1 (s,t))

14 Siggraph 2005, 8/1/ Sketch atlas Properties Charts must cover sphere Overlap substantially with neighbors But not non-adjacent elements Relate to sketch mesh One chart for each vertex, edge, face Vertex: centered, overlaps ½ adjacent edges/faces Edge: covers end points, centroids of adjacent faces Face: covers interior of face Vertex charts Face charts Edge charts

15 Siggraph 2005, 8/1/ Creating embedding Mesh in 1-1 correspondence with sphere Subdivide original 3D mesh and mesh embedded in domain Keeps correspondences Fit chart embedding to corresponding part of subdivision surface

16 Siggraph 2005, 8/1/ Blend and embedding functions Blend: Uniform B-Spline basis function “spun” around origin Produces smallest second derivative variations Embed: C 6 polynomial No problems with end-conditions, skew Continuity: Continuity of blend functions (everything else is C  )

17 Siggraph 2005, 8/1/ Advantages Transition functions are “free”, C  No difficulties with co-cycle condition Can add additional charts anywhere Embedding function provides a natural mechanism for “masking out” original surface Mask out blend functions C k surface pasting with no constraints Re-parameterization is easy Take out old charts and add new ones

18 Siggraph 2005, 8/1/ Disadvantages Need to make/find canonical manifold for each genus May prove difficult for higher dimensions Requires “tweaking” of chart construction to get “nice” overlaps, guaranteed coverage Regularization of embedded sketch mesh Heuristics/optimizations for chart construction Warping function

19 Siggraph 2005, 8/1/ Torus Similar to circle example Repeat in both s and t: [0,2  ) X [0,2  ) Chart is defined by projective transform Care with wrapping (0,0) (2  2  ) X X

20 Siggraph 2005, 8/1/ Torus, associated edges Cut torus open to make a square Two loops (yellow one around, grey one through) Each loop is 2 edges on square Glue edges together Loops meet at a point

21 Siggraph 2005, 8/1/ N-Holed tori Similar to torus – cut open to make a 4N-sided polygon Two loops per hole (one around, one through) Glue two polygon edges to make loop Loops meet at a point Polygon vertices glue to same point Front Back a a b b c c d d

22 Siggraph 2005, 8/1/ Hyperbolic geometry Why is my polygon that funny shape? Need corners of polygon to each have 2  / 4N degrees (so they fill circle when glued together) Tile hyperbolic disk with 4N-sided polygons

23 Siggraph 2005, 8/1/ Hyperbolic geometry Edges are circle arcs; circles meet boundary at right angles Linear fractional transforms Equivalent to matrix operations in Euclidean geometry, e.g., rotate, translate, scale Invertible Chart: Use a Linear fractional transform to map point(s) to origin, then apply warp function Need to ensure we use correct copy in chart function

24 Siggraph 2005, 8/1/ Summary There are some manifolds we use often Sphere, tori, circle, plane, S 3 (quaternions) Construct a general-purpose manifold + atlas + chart creation + transition functions Now can use any tools that operate in R n Use same tools for all topologies Can build charts at any scale, anywhere Not dictated by initial construction/sketch

25 Siggraph 2005, 8/1/ Open questions Boundaries Function discontinuities Changing topologies Establishing correspondences between existing surfaces and canonical manifold Parameterization (next section) Where and how to place charts