ISAMA 2004, Chicago K12 and the Genus-6 Tiffany Lamp Carlo H. Séquin 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
This is Called a Bi-partite Graph K3,4 Alice Bob Pat Harry “Person”-Nodes “Shop”-Nodes
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.
What is “K12” ? • (Unipartite) complete graph with 12 vertices. • Every node connected to every other one ! • In the plane:has lots of crossings…
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.
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 !
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]
My First Model • Find highest-symmetry genus-6 surface, • with “convenient” handles to route edges.
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.
My Model (3) • Now need to place 66 edges: • Use trial and error. • Need a 3D model ! • CAD model much later...
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
Another View of Dyck’s Graph • Difficult to connect up matching nodes !
Towards a 3D Model • Find highest-symmetry genus-3 surface: Klein Surface (tetrahedral frame).
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).
A First Physical Model • Edges of graph should be nice, smooth curves. Quickest way to get a model: Painting a physical object.
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
“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).
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
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 !
K4,4,4 on a Genus-3 Surface LVC on subdivision surface – Graph edges enhanced
Finishing of 3D Color Parts Infiltrate Alkyl Cyanoacrylane Ester = “super-glue” to harden parts and to intensify colors.
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 !
Light Cast by Genus-3 “Tiffany Lamp” Rendered with “Radiance” Ray-Tracer (12 hours)