Voronoi Diagrams for Oriented Spheres Franz Aurenhammer Joint

Voronoi Diagrams for Oriented Spheres Franz Aurenhammer Joint work with M. Peternell H. Pottmann J. Wallner Dagstuhl Seminar Computational Geometry

Voronoi Diagram The classical case …. Size small, easy computation Separators are lines (hyperplanes) Dagstuhl Seminar Computational Geometry 1

Power Diagram Theorem [AI, 1988] Separators are hyperplanes iff the diagram is the power diagram for some set of spheres. Dagstuhl Seminar Computational Geometry 2

Quadratic-form Distance Q(p, q) = (q-p)T · M · (q-p), point sites p, q M nonsingular, k x k T T W. l. o. g. M symmetric: Q(p, q) = ½ (q-p) · (M+M ) · (q-p) Separators are hyperplanes Power diagrams are induced Dagstuhl Seminar Computational Geometry 3

Examples M =I closest-point Voronoi diagram (Eucl. squared) M = -I farthest-point diagram M= ( ) 0 1 1 0 Q(p, q) is twice the area of rectangle T with diagonal pq [CDL] M = diag (1, …, 1, -1) quasi-Euclidean distance Dagstuhl Seminar Computational Geometry 4

Oriented Spheres Points in 3 D Oriented spheres in 2 D Quasi-Eucl. distance d: squared tangent length Principal spheres Dagstuhl Seminar Computational Geometry 5

Motivation Special relativity: Events = points in quasi-Euclidean space (pseudo-metric governed by M) Isometric mappings = Lorentz transformations Value of d(p, q) is a Lorentz invariant < 0 time-like (√|d| = life time) > 0 space-like (√d = Euclidean distance) = 0 light-like Dagstuhl Seminar Computational Geometry 6

Physical Meaning of d Light cone (d=0) separates time domain (d>0) from space domain (d<0) Dagstuhl Seminar Computational Geometry 7

Diagram for d (space ≈ - time) Just a power diagram (for the principal spheres). But: d is not a metric (light-cones) Sidedness may be violated Site extremal region unbounded Dagstuhl Seminar Computational Geometry 8

Variant 1 (time ≈ space) Distance D D = |d(p, q)| Two types of separators Structure determined by light cone arrangement Dagstuhl Seminar Computational Geometry 9

Variant 2 (space driven) Distance Δ ∞ = d(p, q) if ≥ 0 {∞ otherwise Refinement of light cone arrangement Not face-to-face Dagstuhl Seminar Computational Geometry 10

Variant 3 (space driven) Distance Δ 0 = d(p, q) if > 0 {0 otherwise Lives in the complement of the union of time domains Face-to-face Dagstuhl Seminar Computational Geometry 11

What else Associated Delaunay triangulations? Quasi-Euclidean circum-circle…. Dagstuhl Seminar Computational Geometry 12

Thank you Dagstuhl Seminar Computational Geometry 13