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Parameterizing N-holed Tori. Cindy Grimm (Washington Univ. in St. Louis) John Hughes (Brown University). Parameterizing n-holed tori. “Natural” method for parameterizing non-planar topologies Constructive Amenable to spline-like embedding Control points Local control Polynomial. Outline. - PowerPoint PPT Presentation
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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