Voronoi Diagrams and Delaunay Triangulations Generalized spaces and distances

Generalized spaces

Generalized spaces - sphere

Generalized spaces - cone

Generalized spaces – orbits & quotient space Let P denote the Euclidean plane and let G be a discrete group of motions on P: a group of bijections, s.t:

Generalized spaces – orbits & quotient space Definitions: Two points p,p’ ∈ P are equivalent if The equivalence class, [p], of p is called the orbit of p. The quotient space, P/G, consists of all orbits. A connected subset of P that contains a representative out of every orbit called a fundamental domain, if it is convex. Lemma [Ehrlich and Im Hof]: Let p be a point of P that is left fixed only by the unit element of G. Then its Voronoi region VR(p,[p]) is a fundamental domain.

Generalized spaces – Voronoi diagram of quotient space Let S 0 denote a set of representatives of S in a fundamental domain D ⊂ P. Compute the Voronoi diagram V([S 0 ]), skipping all Voronoi edges that separate points of the same orbit. Intersect the resulting structure with the fundamental domain D.

Convex distance functions

Definition: the set of all points q satisfying d(0,q) ≤ 1 is the unit circle. Properties: The value of d C (p,q) does not change if both p and q are translated by the same vector. the triangle inequality: d C (p,r) ≤ d C (p,q)+d C (q,r) d C (p,q)=d C’ (q,p), where C’ denotes the reflected image of C about the origin.

Convex distance functions Definition: Norm p is a function p:V→ ℝ s.t: p(x)≥0 (p(x)=0  x=0) p(λx)=|λ| p(x) p(x+y)≤p(x)+p(y) If the set C is symmetric about the origin then d C is a norm in the plane. Well-known is the family of L p (or: Minkowski) norms, p ≥1, defined by the equation

The L p norms L 1 (q,r) = |q 1 −r 1 |+|q 2 −r 2 | (the Manhattan distance) Unit circle: L ∞ (q,r) = max(|q 1 −r 1 |, |q 2 −r 2 |) Unit circle:

Bisectors under the Manhattan distance (L 1 ) p q p q

Strictly convex distance function Each bisector B(p, q) is a curve homeomorphic to a line. The strict triangle inequality holds. Two circles with respect to d C intersect in at most two points. Two bisectors B(p, q), B(p, r) intersect in at most one point.

Voronoi diagrams under CD function Voronoi regions are in general not convex. Voronoi regions are star-shaped: For each point x ∈ D(p,q), the line segment px is also contained in D(p,q).

Algorithms

Symmetric CD vs. Euclidean distance Symmetric CD is equivalent to the Euclidean distance in the sense that: But:

Symmetric CD vs. Euclidean distance Theorm [Corbalan, Mazon, Recio, and Santos]: If for each set S of at most 4 points the Voronoi diagram V C (S) has the same combinatorial structure as V D (f(S)), for some bijection f of the plane, then f is linear and f(C) = D holds, up to scaling.

Symmetric CD vs. Euclidean distance

CD function in higher dimensions Properties in 3-space: The bisector B(p,q) of two points is a surface homeomorphic to the plane. Two bisector surfaces B(p,q), B(p,r) have an intersection homeomorphic to the line, or empty. Theorem [Icking, Klein, Lê and Ma]: For each n>0 there exist C and 4 points s.t. there are 2n+1 homothetic copies of C containing these points in their boundaries. The Voronoi diagram of n points in 3-space based on the L 1 norm is of complexity Θ(n 2 ), and the diagram in d-space based on L ∞ are of complexity Θ(n ⌈d/2⌉ ).

Metrics

Metric, m, associates with any two points, p and q, a non-negative real number m(p,q). Properties: m(p,q) = 0 ⇔ p = q. m(p,q) = m(q,p). The triangle inequality m(p,r) ≤ m(p,q)+m(q,r) holds.

Metrics - example Suppose there is an air-lift between two points a, b in the plane. Then

Metrics - example Voronoi regions with respect to this metric m: a qb f VR m (a) VR m (q)

Nice metrics Definition: A metric m in the plane is called nice if it enjoys the following properties: A sequence p i converges to p under m iff this holds under the Euclidean distance. For any two points p, r there exists a point q different from p and r s.t. m(p,r)=m(p,q)+m(q,r). For any two points p, q: B m (p,q) is a curve homeomorphic to the line. The intersection of two bisector curves consists of only finitely many connected components.

Nice metrics Lemma [Menger]: Let m be a nice metric in the plane. Then for any two points p, r there exists a path π connecting them, such that for each point q on π: m(p,r)=m(p,q)+m(q,r).

Voronoi diagrams based on nice metrics Lemma (m-star-shaped): Let m be a nice metric. Then each Voronoi region VR m (p,S) is connected: Each m- straight path π from p to some point x ∈ VR m (p,S) is fully contained in VR m (p,S).

Nice metrics – example (Karlsruhe/Moscow)

Algorithms Theorem [Dehne and Klein]: The Voronoi diagram of n point sites under a nice metric in the plane can be constructed within O(nlogn) many steps, using the sweep line approach.

General Voronoi diagrams

Edelsbrunner and Seidel approach Given: S, a set of indices {1,2,…}. A fixed domain, X. For each i ∈ S a real-valued function f i :X→ ℝ. is a hypersurface in the space X ╳ ℝ. The Voronoi diagram is the projection onto X of the lower envelope:

Klein approach – Abstract Voronoi diagram Given: S, a set of indices {1,2,…}. For any two different indices p,q ∈ S, a bisecting curve J(p,q)=J(q,p) homeomorphic to the line. The Voronoi region VR(p,S) is defined as the intersection of the open domains D(p,q).

Admissible system Definition: The system is called admissible iff for each S’ ⊆ S of size at least 3 the following conditions are fulfilled: The Voronoi regions are connected. Each point of the plane lies in a Voronoi region (or on the Voronoi diagram). The intersection of two curves consists of only finitely many components.

Admissible system Lemma: Let ℐ be an admissible system, and suppose that J(p,q) and J(p,r) cross at the point x. Then J(q,r) also passes through x.

Computing diagram of admissible system Divide & conquer ☑ Randomized incremental ☑ Sweep ☒

Questions? Thank you!