1 / 43

ISAMA 2004, Chicago

ISAMA 2004, Chicago. K 12 and the Genus-6 Tiffany Lamp. Carlo H. S é quin and Ling Xiao EECS Computer Science Division University of California, Berkeley. Bob. Alice. Graph-Embedding Problems. Pat. Harry. On a Ringworld (Torus) this is No Problem !. Alice. Bob. Pat.

elvis-morin
Download Presentation

ISAMA 2004, Chicago

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ISAMA 2004, Chicago K12 and the Genus-6 Tiffany Lamp Carlo H. Séquin and Ling Xiao EECS Computer Science Division University of California, Berkeley

  2. Bob Alice Graph-Embedding Problems Pat

  3. Harry On a Ringworld (Torus) this is No Problem ! Alice Bob Pat

  4. This is Called a Bi-partite Graph K3,4 Alice Bob Pat Harry “Person”-Nodes “Shop”-Nodes

  5. 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:= Dyck’s graph • Nodes of the same color are not connected.

  6. What is “K12” ? • (Unipartite) complete graph with 12 vertices. • Every node connected to every other one ! • In the plane:has lots of crossings…

  7. 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.

  8. Not Just Stringing Wires in 3D … • Icosahedron has 12 vertices in a nice symmetrical arrangement; -- let’s just connect those … • But we want graph embedded in a (orientable) surface !

  9. 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 = 2*Genus • 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]

  10. The Connectivity of Bokowski’s Map

  11. Prof. Bokowski’s Goose-Neck Model

  12. Bokowski’s ( Partial ) Virtual Model on a Genus 6 Surface

  13. My First Model • Find highest-symmetry genus-6 surface, • with “convenient” handles to route edges.

  14. 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.

  15. My Model (3) • Now need to place 66 edges: • Use trial and error. • Need a 3D model ! • CAD model much later...

  16. 2nd Problem : K4,4,4 (Dyck’s Map) • 12 nodes (vertices), • but only 48 edges. • E – V – F + 2 = 2*Genus • 48 – 12 – 32 + 2 = 6  Genus = 3

  17. Another View of Dyck’s Graph • Difficult to connect up matching nodes !

  18. Folding It into a Self-intersecting Polyhedron

  19. Towards a 3D Model • Find highest-symmetry genus-3 surface: Klein Surface (tetrahedral frame).

  20. 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).

  21. A First Physical Model • Edges of graph should be nice, smooth curves. Quickest way to get a model: Painting a physical object.

  22. Geodesic Line Between 2 Points • Connecting two given points with the shortest geodesic line on a high-genus surface is an NP-hard problem. T S

  23. “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).

  24. 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

  25. 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 !

  26. K4,4,4 on a Genus-3 Surface LVC on subdivision surface – Graph edges enhanced

  27. K12 on a Genus-6 Surface

  28. 3D Color Printer(Z Corporation)

  29. Cleaning up a 3D Color Part

  30. Finishing of 3D Color Parts Infiltrate Alkyl Cyanoacrylane Ester = “super-glue” to harden parts and to intensify colors.

  31. Genus-6 Regular Map

  32. Genus-6 Regular Map

  33. “Genus-6 Kandinsky”

  34. Manually Over-painted Genus-6 Model

  35. Bokowski’s Genus-6 Surface

  36. Tiffany Lamps (L.C. Tiffany 1848 – 1933)

  37. Tiffany Lamps with Other Shapes ? Globe ? -- or Torus ? Certainly nothing of higher genus !

  38. Back to the Virtual Genus-3 Map Define color panels to be transparent !

  39. A Virtual Genus-3 Tiffany Lamp

  40. Light Cast by Genus-3 “Tiffany Lamp” Rendered with “Radiance” Ray-Tracer (12 hours)

  41. Virtual Genus-6 Map

  42. Virtual Genus-6 Map (shiny metal)

  43. Light Field of Genus-6 Tiffany Lamp

More Related