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CONVEX POLYTOPES. Gleb Kodinets. GALE TRANSFORM. GALE TRANSFORM. בדומה לטרנספורם דואלי טרנספורם גאיל מעביר קונפיגורציה גאומטרית אחד לקונפיגורציה גאומטרית אחרת. המציאו אותו כדי ללמוד יותר את פאונים הקמורים ממימד גבוה יותר טוב. - PowerPoint PPT Presentation

CONVEX POLYTOPES

CONVEX POLYTOPES

Gleb Kodinets11GALE TRANSFORM2GALE TRANSFORM . . , .3Gale Transform45 a Rd5How to67

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Observation. . 9Observation. 10

Observation11

Two ways of probing a configuration

12Two ways of probing a configuration

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171819Lemma20Proof21Proof22Dictionary of the Gale transform23Dictionary of the Gale transform.2425

Signs suffice. 262728

Affine Gale diagrams. 293031

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Easy to check:

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34Proposition (Dictionary of affine Gale diagrams)3536A nonrational polytope. 37Example8-dimensional polytope with 12 vertices that cannot be realized with rational coordinates; that is, no polytope with isomorphic face lattice has all vertex coordinates rational. First one has to become convinced that if 9 distinct points are placed in R2 so that they are not all collinear and there are collinear triples and 4-tuples as is marked by segments in the left drawing below, then not all coordinates of the points can be rational.

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3940Voronoi Diagrams4142

Here is an example of the Voronoi diagram of 2 points in the plane:

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Here is an example of the Voronoi diagram of 3 points in the plane:

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Here is an example of the Voronoi diagram of a point set in the plane:

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ObservationEach region reg(p) is a convex polyhedron with at most |P|-1 facets. Indeed, reg(p) is an intersection of |P| - 1 half-spaces.

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For d = 2, a Voronoi diagram of n points is a subdivision of the plane into n convex polygons (some of them are unbounded). It can be regarded as a drawing of a planar graph (with one vertex at the infinity, say), and hence it has a linear combinatorial complexity: n regions, O(n) vertices, and O(n) edges.Eulers formula: v+f=2+e47Examples of applications. Voronoi diagrams have been reinvented and used in various branches of science. Sometimes the connections are surprising. For instance, in archaeology, Voronoi diagrams help study cultural influences.

48Examples of applications: "Post office problem" or nearest neighbor searching Given a point set P in the plane, we want to construct a data structure that finds the point of P nearest to a given query point x as quickly as possible. This problem arises directly in some practical situations or, more significantly, as a subroutine in more complicated problems. The query can be answered by determining the region of the Voronoi diagram of P containing x. For this problem (point location in a subdivision of the plane), efficient data structures are known.49Robot motion planningConsider a disk-shaped robot in the plane. It should pass among a set P of point obstacles, getting from a given start position to a given target position and touching none of the obstacles.

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Robot motion planningIf such a passage is possible at all, the robot can always walk along the edges of the Voronoi diagram of P, except for the initial and final segments of the tour. This allows one to reduce the robot motion problem to a graph search problem: We define a subgraph of the Voronoi diagram consisting of the edges that are passable for the robot.

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A nice triangulation: the Delaunay triangulation52A nice triangulation: the Delaunay triangulationOne particular triangulation that is usually very good, and provably optimal with respect to several natural criteria, is obtained as the dual graph to the Voronoi diagram of P. Two points of P are connected by an edge if and only if their Voronoi regions share an edge. 53

A nice triangulation: the Delaunay triangulationIf no 4 points of P lie on a common circle then this indeed defines a triangulation, called the Delaunay triangulation of P.The definition extends to points sets in Rd in a straightforward manner.

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Interpolation55Interpolation56

Interpolation57

Relation of Voronoi diagrams to convex polyhedra. 58

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Proposition61

Proof:6263

Corollary. 64

The farthest-point Voronoi diagram.65