Pappus’s Hexagon Theorem in Real Projective Plane

Triangulating the Real Projective Plane

We consider the problem of computing a triangulation of the real projective plane P , given a finite point set P = {p1, p2, . . . , pn} as input. We prove that a triangulation of P always exists if at least six points in P are in general position, i.e., no three of them are collinear. We also design an algorithm for triangulating P if this necessary condition holds. As far as we know, this is t...

A Constructive Real Projective Plane

The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues's Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal's Theorem, poles and polars. The axioms used for the synthetic treatment are constructive versions of the traditional axioms. The analytic construction is used to verify t...

Group of Homography in Real Projective Plane

A convexity theorem for real projective structures

Given a finite collection P of convex n-polytopes in RP (n ≥ 2), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on M is 1. convex if P contains no triangular polytope, and 2. properly convex ...

Straight Line Arrangements in the Real Projective Plane

Let A be an arrangement of n pseudolines in the real projective plane and let p 3 (A) be the number of triangles of A. Grünbaum has proposed the following question. Are there infinitely many simple arrangements of straight lines with p 3 (A) = 1 3 n(n − 1)? In this paper we answer this question affirmatively.