Cindy Grimm

Parameterizing N-holed Tori

Cindy Grimm (Washington Univ. in St. Louis)

John Hughes (Brown University)

Cindy Grimm

Parameterizing n-holed tori

• “Natural” method for parameterizing non-planar topologies

• Constructive

• Amenable to spline-like embedding– Control points– Local control– Polynomial

Cindy Grimm

Outline

• Related work– Patch approach

• Topology

• Related work– Hyperbolic approach

• Manifold approach– Constructive approach to modeling topology

• Embedding

Cindy Grimm

Previous work

• Subdivision surfaces– Constructive method (arbitrary topology)– Induces local parameterization– C1 continuity, higher order harder

• Patches– “Stitch” together n-sided patches

• Requires constraints on control points

Cindy Grimm

Topology

• Building n-holed tori– Associate sides of 4n

polygon

2

01

3

6

5

7 41a

1b

1c

a

1d

b

d

c0

1

65

4

3

2

7

Cindy Grimm

4n-sided polygon

• One loop through hole– a, a-1

• One loop around hole– b, b-1

• Repeat for n holes

1a1b

1c

a

1d

b

d

c0

1

65

4

3

2

7

2

01

36

57 4

Cindy Grimm

4n-sided polygon

• Vertices of polygon become one point on surface– Ordering of edges not same as ordering on

polygon

2

01

36

57 4

1a1b

1c

a

1d

b

d

c0

1

65

4

3

2

7

Cindy Grimm

Hyperbolic disk

• Unit disk with hyperbolic geometry– Sum of triangle angles < 180

• Lines are circle arcs– Circles meet disk perpendicularly

Cindy Grimm

Hyperbolic polygon

• Putting the two together:– Build 4n-sided polygon in hyperbolic disk

• Angles of corners sum to 2

h

r

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

)()( iss

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

dcz

bazz

dc

ba

1

ac

bd

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

)()( iss

Cindy Grimm

Different approach

• Cover the hyperbolic polygon with a manifold– Locally planar parameterization– Transition functions and blends between

parameterizations

02 1

4 3

7

5 6

)2

1,

2

1(

)2

1,

2

1( )

2

1,

2

1(

)2

1,

2

1(

Cindy Grimm

Different approach

• Embed the manifold– Embedding function for each local

parameterization• Splines, RBFs, etc.

– Blend between local embeddings

)()()( cccc c pEpBpE

Cindy Grimm

Roadmap

• Building a manifold– Constructive definition– Choice of charts, transition function

• Embedding function– Local embedding functions– Blend functions

• Tessellation

• User interaction

Cindy Grimm

Manifold definition

• Traditional: Locally Euclidean– Chart: Map from surface

to plane– Induces overlap regions,

transition functions

s

01

21

10

12

12

02

0

Cindy Grimm

Manifold definition

• Constructive definition– Finite set A of non-empty subsets of R2.

Each subset ci is called a chart.– A set of subsets

• Uii=ci

• Empty, union of disjoint subsets.

– Transition functions between subsets• Reflexive• Symmetric• Transitive

iij cU

))(()(

:1 pp

UU

ijij

jiijij

s

01

21

10

12

12

02

0

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

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

Cindy Grimm

Choice of charts

• 2N+2– One interior (unit disk)– One for each edge (unit square)– One “vertex” (unit disk)

02 1

4 37

5 6

)2

1,

2

1(

)2

1,

2

1( )

2

1,

2

1(

)2

1,

2

1(

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

Cindy Grimm

Status

• Structure which is locally planar – Unit disk– Unit square

• Equate points in each chart– Transition functions/overlap regions

• Topology– No geometry

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

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

Accc

Acccc

pB

pEpBpE

))((

))(())(()(

Cindy Grimm

Examples

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– Ck for any k

Cindy Grimm

Tessellation

Edge Inside Vertex

Cindy Grimm

User interface

• Click and drag

Accc

Acccc

pB

pEpBpE

))((

))(())(()(

Accc

Ac jiijjicc

pB

gtbsbpB

pE))((

)()())((

)( ,

vg

g

bbcij

cij

cij

cij

Cindy Grimm

Future work

• Parameterize existing meshes, subdivision surfaces

• Better embeddings– N-sided patches for inside, vertex charts

• Alternative hyperbolic geometries– Klein-Beltrami