ISAMA 2004, Chicago

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ISAMA 2004, Chicago

• K12 and the Genus-6 Tiffany Lamp

Carlo H. Séquin and Ling Xiao EECS Computer
Science Division University of California,
Berkeley
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Graph-Embedding Problems

• Pat

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On a Ringworld (Torus) this is No Problem !
Alice
Bob
Pat
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This is Called a Bi-partite Graph
K3,4
Alice
Bob
Pat
Harry
Person-Nodes
Shop-Nodes
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A Bigger Challenge K4,4,4

• Tripartite graph
• A third set of nodes E.g., access to airport,
  heliport, ship port, railroad station. Everybody
  needs access to those
• Symbolic view Dycks graph
• Nodes of the same color are not connected.

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What is K12 ?

• (Unipartite) complete graph with 12 vertices.
• Every node connected to every other one !
• In the planehas lots of crossings

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Our Challenging Task

• Draw these graphs crossing-free
• onto a surface with lowest possible genus,e.g.,
  a disk with the fewest number of holes
• so that an orientable closed 2-manifold results
• maintaining as much symmetry as possible.

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Not Just Stringing Wires in 3D

• Icosahedron has 12 vertices in a nice symmetrical
  arrangement -- lets just connect those
  
• But we want graph embedded in a (orientable)
  surface !

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Mapping Graph K12 onto a Surface(i.e., an
orientable 2-manifold)

• Draw complete graph with 12 nodes (vertices)
• Graph has 66 edges (border between 2 facets)
• Orientable 2-manifold has 44 triangular facets
• Edges Vertices Faces 2 2Genus
• 66 12 44 2 12 ? Genus 6
• ? Now make a (nice) model of that !
• There are 59 topologically different ways in
  which this can be done ! Altshuler et al. 96

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The Connectivity of Bokowskis Map
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Prof. Bokowskis Goose-Neck Model
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Bokowskis ( Partial ) Virtual Model on a
Genus 6 Surface
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My First Model

• Find highest-symmetry genus-6 surface,
• with convenient handles to route edges.

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My Model (cont.)

• Find suitable locations for twelve nodes
• Maintain symmetry!
• Put nodes at saddle points, because of 11
  outgoing edges, and 11 triangles between them.

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My Model (3)

• Now need to place 66 edges
• Use trial and error.
• Need a 3D model !
• CAD model much later...

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2nd Problem K4,4,4 (Dycks Map)

• 12 nodes (vertices),
• but only 48 edges.
• E V F 2 2Genus
• 48 12 32 2 6 ? Genus 3

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Another View of Dycks Graph

• Difficult to connect up matching nodes !

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Folding It into a Self-intersecting Polyhedron
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Towards a 3D Model

• Find highest-symmetry genus-3 surface? Klein
  Surface (tetrahedral frame).

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Find Locations for Nodes

• Actually harder than in previous example, not
  all nodes connected to one another. (Every node
  has 3 that it is not connected to.)
• Place them so that themissing edges do not
  break the symmetry
• ? Inside and outside on each tetra-arm.
• Do not connect the nodes that lie on thesame
  symmetry axis(same color)(or this one).

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A First Physical Model

• Edges of graph should be nice, smooth curves.

Quickest way to get a model ? Painting a
physical object.
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Geodesic Line Between 2 Points
T
S

• Connecting two given points with the shortest
  geodesic line on a high-genus surface is an
  NP-hard problem.

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Pseudo Geodesics

• Need more control than geodesics can offer.
• Want to space the departing curves from a vertex
  more evenly, avoid very acute angles.
• Need control over starting and ending tangent
  directions (like Hermite spline).

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LVC Curves (instead of MVC)

• Curves with linearly varying curvaturehave two
  degrees of freedom kA kB,
• Allows to set two additional parameters,i.e.,
  the start / ending tangent directions.

CURVATURE
kB
ARC-LENGTH
kA
B
A
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Path-Optimization Towards LVC

• Start with an approximate path from S to T.
• Locally move edge crossing points ( C ) so as to
  even out variation of curvature

S
C
V
C
T

• For subdivision surfaces refine surface and LVC
  path jointly !

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K4,4,4 on a Genus-3 Surface

• LVC on subdivision surface Graph edges
  enhanced

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K12 on a Genus-6 Surface
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3D Color Printer (Z Corporation)
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Cleaning up a 3D Color Part
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Finishing of 3D Color Parts

• Infiltrate Alkyl Cyanoacrylane Ester
  super-glue to harden parts and to intensify
  colors.

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Genus-6 Regular Map
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Genus-6 Regular Map
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Genus-6 Kandinsky
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Manually Over-painted Genus-6 Model
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Bokowskis Genus-6 Surface
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Tiffany Lamps (L.C. Tiffany 1848 1933)
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Tiffany Lamps with Other Shapes ?
Globe ? -- or Torus ? Certainly nothing of
higher genus !
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Back to the Virtual Genus-3 Map
Define color panels to be transparent !
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A Virtual Genus-3 Tiffany Lamp
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Light Cast by Genus-3 Tiffany Lamp

• Rendered with Radiance Ray-Tracer (12 hours)

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Virtual Genus-6 Map
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Virtual Genus-6 Map (shiny metal)
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Light Field of Genus-6 Tiffany Lamp