1 Adding charts anywhere Assume a cow is a sphere Cindy Grimm and John Hughes, “Parameterizing...

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Adding charts anywhereAssume 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, 8/1/2005www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

Motivation

•Surface modeling: Adding more degrees of freedom• Don’t want to be restricted to existing charts/dof

•Computers support vectors (points in Rn)• Functions on Rn

• Splines, gradient descent algorithms• Define functions directly on sphere?

• BRDFs, environment maps


Observation

•We can define lots of meshes that are spheres• Which one’s the “right” one?• How do you compare two sphere meshes?


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 Rn to domain)• Movement in domain

• Barycentric coordinates, linear interpolation


Once we have a domain…

•Add charts. Anywhere.• Not restricted by existing atlas/construction

•Questions to answer:• Chart specification?• Chart function form?


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


Outline

• Spherical domain• Defining• Adding charts• Building embedding function (surface

• Torus, n-holed tori domains• Torus: Tiled plane

• N-holed: Tiled plane, sort-of


Atlas from mesh

Embedded sketch Vertex chartsSketch Surface

Embed mesh in domain• 1-1 correspondence • Specifies both chart locations and chart embeddings


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


Representing a sphere

• Representing points• x2+y2+z2 -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)


Charts

• Stereographic projection• Sketch mesh charts• Circles to circles

• Requires rotation to center at arbitrary point

• Both C

Images: mathworld.wolfram.com


Charts, adjusting

•Raw projections MD don’t provide much control

• Map to consistent chart shape (e.g., unit disk)

•Add additional “warp” function MW

• Projective transform (scale, translate, rotate)

• Still C

DM

CU1DM

wM

1wM

c


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))


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


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


Blend and embedding functions

•Blend: Uniform B-Spline basis function “spun” around origin

• Produces smallest second derivative variations•Embed: C6 polynomial

• No problems with end-conditions, skew

•Continuity: Continuity of blend functions (everything else is C)


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

• Ck surface pasting with no constraints•Re-parameterization is easy

• Take out old charts and add new ones


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


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)

(22)

X

X


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


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

aab

b

ccd

d


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


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


Summary

•There are some manifolds we use often• Sphere, tori, circle, plane, S3 (quaternions)

•Construct a general-purpose manifold + atlas + chart creation + transition functions

• Now can use any tools that operate in Rn • Use same tools for all topologies

• Can build charts at any scale, anywhere• Not dictated by initial construction/sketch


Open questions

•Boundaries•Function discontinuities•Changing topologies•Establishing correspondences between existing surfaces and canonical manifold

• Parameterization (next section)•Where and how to place charts

1 Adding charts anywhere Assume a cow is a sphere Cindy Grimm and John Hughes, “Parameterizing...

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