Presentation on theme: "ISAMA 2004, Chicago K 12 and the Genus-6 Tiffany Lamp Carlo H. Séquin and Ling Xiao EECS Computer Science Division University of California, Berkeley."— Presentation transcript:
ISAMA 2004, Chicago K 12 and the Genus-6 Tiffany Lamp Carlo H. Séquin and Ling Xiao EECS Computer Science Division University of California, Berkeley
Graph-Embedding Problems Bob Alice Pat
On a Ringworld (Torus) this is No Problem ! Bob Alice Pat Harry
This is Called a Bi-partite Graph Bob Alice Pat Harry Person-Nodes Shop-Nodes K 3,4
A Bigger Challenge : K 4,4,4 u Tripartite graph u A third set of nodes: E.g., access to airport, heliport, ship port, railroad station. Everybody needs access to those… u Symbolic view: = Dycks graph u Nodes of the same color are not connected.
What is K 12 ? u (Unipartite) complete graph with 12 vertices. u Every node connected to every other one ! u In the plane: has lots of crossings…
Our Challenging Task Draw these graphs crossing-free l onto a surface with lowest possible genus, e.g., a disk with the fewest number of holes; l so that an orientable closed 2-manifold results; l maintaining as much symmetry as possible.
u Icosahedron has 12 vertices in a nice symmetrical arrangement; -- lets just connect those … u But we want graph embedded in a (orientable) surface ! Not Just Stringing Wires in 3D …
Mapping Graph K 12 onto a Surface (i.e., an orientable 2-manifold) u Draw complete graph with 12 nodes (vertices) u Graph has 66 edges (=border between 2 facets) u Orientable 2-manifold has 44 triangular facets u # Edges – # Vertices – # Faces + 2 = 2*Genus u 66 – 12 – = 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]
The Connectivity of Bokowskis Map
Prof. Bokowskis Goose-Neck Model
Bokowskis ( Partial ) Virtual Model on a Genus 6 Surface
My First Model u Find highest-symmetry genus-6 surface, u with convenient handles to route edges.
My Model (cont.) u Find suitable locations for twelve nodes: u Maintain symmetry! u Put nodes at saddle points, because of 11 outgoing edges, and 11 triangles between them.
My Model (3) u Now need to place 66 edges: u Use trial and error. u Need a 3D model ! u CAD model much later...
2 nd Problem : K 4,4,4 (Dycks Map) u 12 nodes (vertices), u but only 48 edges. u E – V – F + 2 = 2*Genus u 48 – 12 – = 6 Genus = 3
Another View of Dycks Graph u Difficult to connect up matching nodes !
Folding It into a Self-intersecting Polyhedron
Towards a 3D Model u Find highest-symmetry genus-3 surface: Klein Surface (tetrahedral frame).
Find Locations for Nodes u Actually harder than in previous example, not all nodes connected to one another. (Every node has 3 that it is not connected to.) u Place them so that the missing edges do not break the symmetry: u Inside and outside on each tetra-arm. u Do not connect the nodes that lie on the same symmetry axis (same color) (or this one).
A First Physical Model u Edges of graph should be nice, smooth curves. Quickest way to get a model: Painting a physical object.
Geodesic Line Between 2 Points u Connecting two given points with the shortest geodesic line on a high-genus surface is an NP-hard problem. T S
Pseudo Geodesics u Need more control than geodesics can offer. u Want to space the departing curves from a vertex more evenly, avoid very acute angles. u Need control over starting and ending tangent directions (like Hermite spline).
LVC Curves (instead of MVC) u Curves with linearly varying curvature have two degrees of freedom: k A k B, u Allows to set two additional parameters, i.e., the start / ending tangent directions. A B CURVATURE kAkA kBkB ARC-LENGTH
Path-Optimization Towards LVC u Start with an approximate path from S to T. u Locally move edge crossing points ( C ) so as to even out variation of curvature: T C S C V u For subdivision surfaces: refine surface and LVC path jointly !
K 4,4,4 on a Genus-3 Surface LVC on subdivision surface – Graph edges enhanced
K 12 on a Genus-6 Surface
3D Color Printer (Z Corporation)
Cleaning up a 3D Color Part
Finishing of 3D Color Parts Infiltrate Alkyl Cyanoacrylane Ester = super-glue to harden parts and to intensify colors.
Genus-6 Regular Map
Manually Over-painted Genus-6 Model
Bokowskis Genus-6 Surface
Tiffany Lamps (L.C. Tiffany 1848 – 1933)
Tiffany Lamps with Other Shapes ? Globe ? -- or Torus ? Certainly nothing of higher genus !
Back to the Virtual Genus-3 Map Define color panels to be transparent !
A Virtual Genus-3 Tiffany Lamp
Light Cast by Genus-3 Tiffany Lamp Rendered with Radiance Ray-Tracer (12 hours)