1

Spherical manifolds for hierarchical surface modeling

Cindy Grimm

2Graphite 2005, 12/1/2005

Goal

•Organic, free-form•Adaptive

• Can add arms anywhere•Hierarchical

• Move arm, hand moves•Ck analytic surface•Build in pieces

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Overview

• Building an atlas for a spherical manifold• Embedding the atlas

• Surface modeling• Sketch mesh • Adaptive editing

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Traditional atlas definition

Given: Sphere (manifold) M

Construct: Atlas A•Chart

• Region Uc in M (open disk)

• Region c in R2 (open disk)

• Function c taking Uc to c

• Inverse

•Atlas is collection of charts• Every point in M in at least

one chart• Overlap regions

M

01

1

10

0

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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

1222 zyx

All points such that…

pq

(1-t)p + tq

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Chart on a sphere

Chart specification:

•Center and radius on sphere Uc

•Range c = unit disk

•Simplest form for c

• Project from sphere to plane• (stereographic)• Adjust with projective map• Affine

1

c

-1

wM 1wM

DM 1DM

Uc

)),,(( zyxMM DWc

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Defining an atlas

•Define overlaps computationally

•Point in chart: evaluate c

•Coverage on sphere (Uc domain of chart)

• Define in reverse as c-1=MD

-1(MW-1(D))

• D becomes ellipse after warp, ellipsoidal on sphere

• Can bound with cone normal

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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

Accc

ccc

pB

pBpB

''' ))((ˆ

))((ˆ)(

cB

Normalized blend function

cB̂

Proto blend function

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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

Ac

cccc pEpBpE ))(())(()( Embed

Blend

))(( 00 pE

))(( 11 pE

))(( 00 pB

))(( 11 pB

)( pEMap each chart

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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)

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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

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Edge and vertex

•Edge: cover edge, extend to mid-point of adjacent faces•Vertex: Cover adjacent edge mid points, face centers

Edge

VertexEdge charts

Vertex charts

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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

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Summary

•CK analytic surface approximating subdivision surface•Real time editing

• Works for other closed topologies• Parameterization using manifolds, Cindy Grimm,

International Journal of Shape modeling 2004

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Hierarchical editing

•Override surface in an area•Add arms, legs

• User draws on surface• Smooth blend• No geometry constraints

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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

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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

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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

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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

))(())(1()(ˆ pBppB ccjijc

))(()()(ˆ pBppB cciic

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Summary

•Flexible modeling paradigm• No knot lines, geometry constraints

•Not limited to subdivision surfaces• Alternative editing techniques?