Voronoi Diagrams for Oriented Spheres

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

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

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

Quadratic-form Distance T Q(p,q) = (q-p) · 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 Computational Geometry

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

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

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 Computational Geometry

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

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 Computational Geometry

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

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

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

What else AssociatedDelaunay triangulations? Quasi-Euclidean circum-circle…. Computational Geometry

Thank you Computational Geometry

Voronoi Diagrams for Oriented Spheres