We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein's non-conformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. Therefore our method simply consists in computing an equivalent clipped power diagram followed by a mapping transformation depending on the selected representation of the hyperbolic space (e.g., Poincar\'e conformal disk or upper-plane representations). We discuss on extensions of this approach to weighted and $k$-order diagrams, and describe their dual triangulations. Finally, we consider two useful primitives on the hyperbolic Voronoi diagrams for designing tailored user interfaces of an image catalog browsing application in the hyperbolic disk: (1) finding nearest neighbors, and (2) computing smallest enclosing balls.
Deep Dive into Hyperbolic Voronoi diagrams made easy.
We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein’s non-conformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. Therefore our method simply consists in computing an equivalent clipped power diagram followed by a mapping transformation depending on the selected representation of the hyperbolic space (e.g., Poincar'e conformal disk or upper-plane representations). We discuss on extensions of this approach to weighted and $k$-order diagrams, and describe their dual triangulations. Finally, we consider two useful primitives on the hyperbolic Voronoi diagrams for designing tailored user interfaces of an image catalog browsing application in the hyperbolic disk: (1) finding nearest neighbors, and (2) computing smallest enclosing balls.
The birth and spread of science was originally initiated by geometry, the science of (Earth) measurements. Euclid (300 BC) laid the foundations of geometry in his masterpiece book series Elements that still remain the basis of mathematics nowadays, two millenia later. Euclidean geometry was assumed to be the only consistent geometry until the nineteenth century. Eventually, failing to prove Euclid's fifth postulate (known as the parallel postulate) from the others opened the door to abstract geometries. The first two abstract geometries that were historically called imaginary geometries are the hyperbolic geometry and the spherical geometry. The essential difference between Euclidean and these non-Euclidean geometries is the nature of parallel lines. While in Euclidean geometry there is a unique line passing through a given point and parallel to another line, there can be infinitely many in the hyperbolic geometry [1] and none in spherical/elliptical geometries. In this paper, we focus on characterizing and building efficiently the Voronoi diagram in hyperbolic geometry.
The Voronoi diagram [2] of a finite point set partitions the space in cells according to proximity relations induced by a distance function d. The structures of Voronoi diagrams appear in various fields such as crystallography, astronomy and biology. Let P = {p 1 , …, p n } be a set of n d-dimensional vector points of space X . Then the Voronoi diagram of P is defined by the collection of proximal regions vor(p i )’s, called Voronoi cells, such that vor
The Voronoi diagram in hyperbolic geometry has already been partially studied; Onishi and Takayama [3] investigated the hyperbolic Voronoi diagram construction in the Poincaré upper half-plane. Onishi and Itoh [4] further extended the Voronoi diagram in Hadamard manifold that are simply connected complete manifold with non-positive curvature. Boissonnat et al. [5] considered the hyperbolic Voronoi diagram in the Poincaré conformal upper plane and the Poincaré conformal disk. 3 Boissonnat and Yvinec described in their computational geometry textbook the hyperbolic Voronoi diagram in the Poincaré d-dimensional half-space H d + [2], pp. 449-454. They do not introduce explicitly the hyperbolic distance but rather show how to answer proximity queries (e.g., whether B or C is closer to A) using the notion of pencil of spheres. They deduced that the complexity of the hyperbolic Voronoi diagram is O(n log n + n d 2 ). Nilforoushan and Mohades [6] concentrated on the hyperbolic Voronoi diagram in the Poincaré 2D disk for which they report an incremental quadratic algorithm, and characterize geometrically the orthogonality of bisectors with geodesics. Note that by using the graphics processor unit (GPU), it is easy to rasterize interactively Voronoi diagrams [7] for any arbitrary distance function, including the distance of hyperbolic geometry.
In this paper, we revisit these work by considering the Klein projective disk model. We showed in section 2 that bisectors are hyperplanes, implying that the Voronoi diagram is affine and therefore can be easily constructed from an equivalent power diagram. We report the one-to-one conversion formula to change the representation of the hyperbolic geometry: Klein or Poincaré disk models and Poincaré upper half plane model. We further characterize the weighted and k-order diagrams in the Klein disk model and explain the dual hyperbolic Delaunay triangulation. In section 3, we present an image browsing application in the hyperbolic disk that requires two basic user interface selection operations that are solved by means of nearest neighbor queries and by finding the smallest enclosing ball in hyperbolic geometry.
2 Hyperbolic Voronoi diagrams from power diagrams
The Klein model (or Beltrami-Klein model [1]) uses the interior of a unit circle for fully representing the hyperbolic plane. The Klein model is also known as the projective disk model where lines are depicted by chords of the circle (e.g., line segments joining any two points of the circle). Although the Klein model offers a simple visualization of geodesics as line segments, it has the disadvantage of being a non-conformal representation: That is, angles of the hyperbolic plane are not preserved, and get distorted in this model. (That explains why the Poincaré conformal disk representation is often preferred.) The distance between any two points p and q of the Klein disk with Euclidean coordinates (||p|| < 1 and ||q|| < 1) is computed as
where arccosh x = log(x + √ x 2 -1).
Bisectors are core primitives to characterize and compute Voronoi diagrams.
The bisector B of two points p and q with respect to a given distance function D is defined as the loci of the points c that are at the same distance of to p and q: B : {c | D(p, c) = D(q, c)}. Let us derive from the distance function of Eq. 1 the equation of the bisector in Klein’s disc model of hyperbolic geometry:
Thus the bisector is a hyperplan
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
