On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution)

On Balls in a Hilbert Polygonal Geometry Frank Nielsen1 and Laëtitia Shao2 1 2 École Polytechnique, LIX, Palaiseau, France École Polytechnique, Palaiseau, France Abstract Hilbert geometry is a metric geometry that extends the hyperbolic Cayley-Klein geometry. In this video, we explain the shape of balls and their properties in a convex polygonal Hilbert geometry. First, we study the combinatorial properties of Hilbert balls, showing that the shapes of Hilbert polygonal balls depend both on the center location and on the complexity of the Hilbert domain but not on their radii. We give an explicit description of the Hilbert ball for any given center and radius. We then study the intersection of two Hilbert balls. In particular, we consider the cases of empty intersection and internal/external tangencies. 1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling Keywords and phrases Projective geometry, Hilbert geometry, balls Digital Object Identifier 10.4230/LIPIcs.SoCG.2017.67 Category Multimedia Contribution 1 Introduction: Hilbert geometry Hilbert geometry is a projective geometry relying on the properties of the cross-ratio: I Definition 1 (Cross-ratio). For four collinear points a, b, c, d the cross ratio is defined as follows: (a, b; c, d) = kackkbdk kadkkbck (1) The cross-ratio is an invariant measure under perspective transformation: I Property 2 (Projective invariance of the cross-ratio). Given four points a, b, c, d and A, B, C, D their images through a projective transformation, (a, b; c, d) = (A, B; C, D). [5] In a Hilbert geometry, the distance between two points is defined using the cross-ratio as follows: I Definition 3 (Hilbert distance). A Hilbert distance is defined in the interior of a convex bounded domain C. Given two distinct points, a and b of the domain, the distance is defined as follows: dHG (a, b) = log((a, b; A, B)) (2) where (a, b; A, B) is the cross-ratio where A and B denote the intersection points of line (a, b) with the domain. By definition, dHG (x, x) = 0 for all x ∈ C. © Frank Nielsen and Laëtitia Shao; licensed under Creative Commons License CC-BY 33rd International Symposium on Computational Geometry (SoCG 2017). Editors: Boris Aronov and Matthew J. Katz; Article No. 67; pp. 67:1–67:4 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 67:2 On Balls in a Hilbert Polygonal Geometry Figure 1 Left: In blue, two Hilbert balls in a circular domain. Right: In blue, three Hilbert balls in a polygonal convex domain. I Property 4 (Properties of the Hilbert distance). Given two points a and b. The Hilbert distance is a signed distance: dHG (a, b) = −dHG (b, a). dHG (a, a) = 0 (law of the indiscernibles). When a is on the boundary of the convex, ∀b ∈ C, dHG (a, b) = ∞. |dHG | respects the triangular inequality and therefore |dHG | is a metric distance [1]. A key property in Hilbert geometry is that shortest-path geodesics are straight lines. The Klein disk representation of hyperbolic geometry is an example of Hilbert geometry for the unit disk (convex and smooth) domain. In this work, we consider convex polygonal Hilbert geometries, that is, Hilbert geometries defined on a convex polygonal domain. C now refers to a convex polygon with s vertices: e1 , ..es . The distance between two points p and q in this domain is noted dC (p, q) . The ball of radius r and center c is denoted by B(c, r). The sphere is denoted by S(c, r). See [4] for an application of Hilbert geometry to clustering in the open probability simplex. 2 Combinatorial properties of Hilbert balls In Klein ball hyperbolic geometry or Cayley-Klein hyperbolic geometry, the balls have the shape of (Euclidean) Mahalanobis balls with displaced centers, see [2, 3]. To contrast with this smooth shape representation of balls, let us observe that when the domain is a convex polygon, the shapes of Hilbert balls are (Euclidean) polygons. I Definition 5 (Rays). Given a center point c in the domain, line (c, ei ), i ∈ [s] = {1, . . . , s} is a ray. I Lemma 6 (Description of a Hilbert ball). B(c, r) is a Euclidean polygon with at most 2s edges and at least s edges. Each vertex of B(c, r) belongs to a ray. Proof. We first partition the polygonal domain with s to 2s triangles, by tracing rays (c, ei ), i ∈ [s]. We will show that each triangle induces a linear edge of the Hilbert ball. We consider a pair of triangles (A, B, c) and (c, C, D) such that A, c, D and B, c, C are respectively collinear. Let P ∈ [A, c] ∩ B(c, r) and O = (A, B) ∩ (C, D), we will show that line (O, P ) clipped to the triangle (A, B, c) is an edge of B(c, r). Let U be a point on the clipped line, and M, N the intersections points of line (U c) with the domain such that M ∈ [A, B] and N ∈ [C, D]. Then M, U, c, N and A, P, c, D are related by the same projective transformation. Using the invariance property of the cross-ratio, we conclude that dC (c, P ) = dC (c, U ) = r. Thus, we proved Lemma 6. It is remarkable that depending on the position of the center, the number of triangles (and hence the complexity of the ball) varies. J F. Nielsen and L. Shao 67:3 O C A N P c U M Q l2 D B l4 l1 l3 Figure 2 Left: Configuration for proof 2 (see text). Right: Varying number of rays in a square domain depending on the position of the center of the ball. Figure 3 Left: Configuration for proof of Lemma 8 when Ei = Ek . Right: Configuration for proof of Lemma 8 when all edges are distinct. I Definition 7. Given an edge [P, Q] of a Hilbert ball that belongs to a pair of triangles (A, B, c) and (c, D, E), we say that [P, Q] is induced by edges Ei and Ej of the domain, if [A, B] ⊂ Ei and [D, E] ⊂ Ej . I Lemma 8 (Shape invariance with varying radius). For c a fixed center point, and r a varying radius, B(c, r) has the same number of edges. Proof. Let [P, Q] and [Q, R] be two adjacent edges of a Hilbert ball such that Ei , Ej induces [P, Q] and Ek , El induces [Q, R]. We show that P, Q, R cannot be collinear. We note M the intersection points of the lines supported by Ei and Ej and N the intersection points of the lines supported by Ek and El . According to the previous proof, P, Q, M and Q, R, N are respectively collinear. If Ei , Ej , Ek , El are distinct edges, because [P, Q] and [Q, R] are adjacent, we can assume without loss of generality that Ei is adjacent to Ek and Ej is adjacent to El . If P, Q, R are collinear, then Ei = Ek or Ej = El , which contradicts the previous assumption. Otherwise, we can assume that Ei = Ek . In this case, if P, Q, R are collinear, then they belong to line (M, N ) ⊂ Ei . Which is impossible unless r = ∞. Therefore, as the radius varies but stay finite, the number of edges remains constant. See Figure 3 for a visualization of the proof. For infinite radius, all balls fully cover the polygonal domain. J I Lemma 9 (Shape invariance in a simplex domain). In a simplex domain ∆, Hi (...truncated)