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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). Power Diagram.

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Voronoi Diagrams for Oriented Spheres

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

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

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

  4. 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

  5. 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

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

  7. 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

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

  9. 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

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

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

  12. 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

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

  14. Thank you Computational Geometry

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