We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general position, i.e., no three of them are collinear. We also design an algorithm for triangulating P2 if this necessary condition holds. As far as we know, this is the first computational result on the real projective plane.
Deep Dive into Triangulating the Real Projective Plane.
We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,…, pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general position, i.e., no three of them are collinear. We also design an algorithm for triangulating P2 if this necessary condition holds. As far as we know, this is the first computational result on the real projective plane.
Résumé : Nous considérons le calcul de la triangulation d'un ensemble fini de points P = {p 1 , p 2 , . . . , p n } dans le plan projectif. Nous démontrons que la triangulation existe toujours dès lors qu'au moins six points de P sont en position générale, c'est-à-dire que trois d'entre eux ne sont jamais alignés. Nous proposons également un algorithme pour trianguler P 2 si cette condition nécessaire est remplie. À notre connaissance, c'est le premier résultat algorithmique connu pour le plan projectif réel.
Mots-clés : Géométrie algorithmique, triangulation, complexe simplicial, géométrie projective, algorithme
The real projective plane P 2 is in one-to-one correspondence with the set of lines of the vector space R 3 . Formally, P 2 is the quotient P 2 = R 3 -{0} / ∼ where the equivalence relation ∼ is defined as follows: for two points p and p ′ of P 2 , p ∼ p ′ if p = λp ′ for some λ ∈ R-{0}.
Triangulations of the real projective plane P 2 have been studied quite well in the past, though mainly from a graph-theoretic perspective. A contraction of and edge e in a map M removes e and identifies its two endpoints, if the graph obtained by this operation is simple. M is irreducible if none of its edges can be contracted. Barnette [1] proved that the real projective plane admits exactly two irreducible triangulations, which are the complete graph K 6 with six vertices and K 4 + K 3 (i.e., the quadrangulation by K 4 with each face subdivided by a single vertex), which are shown in Figure 1. Note that these figures are just graphs, i.e. the horizontal and vertical lines do not imply collinearity of the points. A diagonal flip is an operation which replaces an edge e in the quadrilateral D formed by two faces sharing e with another diagonal of D (see Figure 2). If the resulting graph is not simple, then we do not apply it. Wagner [18] proved that any two triangulations on the plane with the same number of vertices can be transformed into each other by a sequence of diagonal flips, up to isotopy. This result has been extended to the torus [5], the real projective plane and the Klein bottle [16]. Moreover, Negami has proved that for any closed surface F 2 , there exists a positive integer N (F 2 ) such that any two triangulations G and G ′ on F 2 with |V (G)| = |V (G ′ )| ≥ N (F 2 ) can be transformed into each other by a sequence of diagonal flips, up to homeomorphism [14]. Mori and Nakamoto [11] gave a linear upper bound of (8n -26) on the number of diagonal flips needed to transform one triangulation of P 2 into another, up to isotopy. There are many papers concerning with diagonal flips in triangulations, see [15,7] for more references. In this paper, we address a different problem, which consists in computing a triangulation of the real projective plane, given a finite point set P = {p 1 , p 2 , . . . , p n } as input.
Definition 1.1 Let us recall background definitions here. More extensive definitions are given for instance in [19,9]. M. Aanjaneya & M. Teillaud An (abstract) simplicial complex is a set K together with a collection S of subsets of K called (abstract) simplices such that:
For all v ∈ K, {v} ∈ S. The sets {v} are called the vertices of K.
If τ ⊆ σ ∈ S, then τ ∈ S.
Note that the property that σ, σ ′ ∈ K ⇒ σ ∩ σ ′ ≤ σ, σ ′ can be deduced from this.
σ is a k-simplex if the number of its vertices is k + 1. If τ ⊂ σ, τ is called a face of σ.
A triangulation of a topological space X is a simplicial complex K such that the union of its simplices is homeomorphic to X.
All algorithms known to compute a triangulation of a set of points in the Euclidean plane use the orientation of the space as a fundamental prerequisite. The projective plane is not orientable, thus none of these known algorithm can extend to P 2 .
We will always represent P 2 by the sphere model where a point p is same as its diametrically opposite “copy” (as shown in Figure 3(a)). We will refer to this sphere as the projective sphere. A triangulation of the real projective plane P 2 is a simplicial complex such that each face is bounded by a 3-cycle, and each edge can be seen as a greater arc on the projective sphere. We will also sometimes refer to a triangulation of the projective plane as a projective triangulation.
The sphere model of P 2 . (b) △pqr separated from its “copy” by a distinguishing plane in R 3 .
Stolfi [17] had described a computational model for geometric computations: the oriented projective plane, where a point p and its diametrically opposite “copy” on the projective sphere are treated as two different points. In this model, two diametrically opposite triangles are considered as different, so, the computed triangulations of the oriented projective plane are actually not triangulations of P 2 . Identifying in practice a triangle and its opposite in some data-structure is not straightforward. Let us also mention that the oriented projective model can be pretty costly because it involves the duplic
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