Detecting Induced Incidences in the Projective Plane

Finding forced collinearity relationship can be usefull in several domains of algorithmic geometry. In particular, in the GCS framework, it can help to detect bad-constrained systems. This paper deals with a simple framework where only incidence and collinarities are considered. We propose two methods to prove that a collinearity is a logical consequence of other specified collinearities. 1 GCS and dependency Solvers of geometric constraints often assume that geometric constraints (equations) are independent. The arising problem of detecting dependences between geometric constraints can be approached in many ways [7, 5]. Since the incidence dependences are the most basic in geometry, this paper focuses on the following question: in the projective plane (for simplicity), and given only collinearities constraints between points, is it possible to detect induced or forced (non initially specified) collinearities, or concurrencies between lines? Such forced incidences are due to the pseudo transitivity of the collinearity relationship, but also to Désargues’ or Pappus’theorems. Note that, if we abandon the hypothesis that only incidence constraints are given, the Euler’s line and the Simpson’s line give other examples of forced collinearity relationships. In particular, we suggest some combinatorial methods for proving incidence theorems in the projective plane. Actually, these methods are not very efficient, but we hope they are a first step towards more effective algorithms which can be used in order to make the geometric solvers more robust. The rest of the paper is organized as follows. In section 2, we recall some fundamental facts of projective geometry. Section 3 presents a method based on the hexamys property. In section 4, we explain another method based on the computation of matroids compatible with the collinearity constraints given by the user. We conclude in section 5. 2 Projective geometries First, let us recall the classical definition of a projective geometry Definition 1. A projective geometry is a set S of points and a collection of subsets of S, the set of lines, subject to these axioms: (P1) each pair A, B of distinct points is contained in a unique line which is denoted (AB), (P2) if A, B, C and D are distinct points for which (AB) ∩ (CD) 6= ∅, then (AC) ∩ (BD) 6= ∅, and (P3) each line contains at least three points. This statement can be specialized into the following definition of a projective plane. Definition 2. A projective plane is a set S of points and a collection of subsets of S, the set of lines, subject to these axioms: (P1) each pair A, B of distinct points is contained in a unique line which is denoted (AB), (P2) each pair of distinct lines intersects in a single point, and (P3) each line contains at least three points. If, for some n > 2, n points A1, . . . An are on the same line, one says that there are collinear. This is denoted [A1, . . . , An] It follows directly from the definition that [A,B,C] and [A,B,D] implies that [A,B,C,D]. This fact is true in every projective plane and it constitutes the simplest example of induced incidence. There is a lot of example of projective geometries, in particular, finite projective geometries which are still very studied. Real or complex geometric projective spaces of dimension n, that is projective geometries arising from R or C, are other usual examples. But, not all projective planes arise from a field. More precisely, the important following theorems hold. Theorem 1. A projective plane is isomorphic to a projective plane arising from a division ring if and only if it satisfies the Désargues’ property. Such a plane is called an arguesian plane. A projective plane is isomorphic to a projective plane arising from a field if and only if it satisfies the Pappus’ property. Any projective geometry of dimension greater than 2 is isomorphic to a projective geometry arising from a division ring (and the Désargues’ property is true). The Désargues’ property and the Pappus’ property are usually known as the Désargues’ theorem and the Pappus’ theorem since they are true in the real projective plane. Fig. 1 and Fig. 2 recall the statement of these theorems.