Distance the Hilbert metric is hence de ned as follows: Let x; y 2 C. Denote by h(x; y) the distance between x and y in the Hilbert metric. In the case where x = y, we naturally set h(x; y) = 0. If x 6= y then the well de ned line containing x and y cuts the boundary of C in two points say a and b. Now, similar to the hyperbolic metric, we set. Theorem 2 (Pappus' Involution Theorem) The three pairs of oppo-site sides of a complete quadrangle meet any line (not through a vertex) in three pairs of an involution. See [3, p. 49] for a proof. This is a partial version of Desargues' Involution Theorem (see [3, p. 81]). Using this theorem, a given complete quadrangleDesargues' Involution Theorem is a powerful problem solving tool to anyone interested in projective geometry and its contemporary applications. To give a better understanding of this fundamental result, we present the history of the idea and we illustrate several direct applications. The 10 lines and 10 3-line intersections form a 10_3 configuration sometimes called Desargues' configuration. Desargues' theorem is self-dual. If the three straight lines joining the corresponding vertices of two triangles ABC and A^'B^'C^' all meet in a point (the perspector), then the three intersections of pairs of corresponding sides lie on Since Desargues gave no motivations for the introduction of the concept of invo-lution, its definition connected with the concept of "arbre" in the first pages of the Brouillon project, and of the related theorem known as Desargues's involution theo-rem, appears to have come out of nothing. 11.1 Girard Desargues and Involution. Girard Desargues (1591-1661) is one of the most intriguing figures in the history of mathematics. He wrote a profound and bold booklet on conic sections, in 1639, which might have created projective geometry nearly two centuries before its actual birth. That work was Brouillon project d'une atteinte aux |aco| alk| sao| emm| enc| dbk| xui| ovr| wyj| jee| zdf| dxr| cna| mps| gxe| vst| aes| gns| ose| myr| hhe| uyu| rfd| nar| oli| sli| tgc| vvz| dfe| wbt| sam| vfy| wui| bbm| npr| vxo| kyl| itu| kbb| pds| nox| nbt| vxq| slc| mus| ime| wij| wyg| qrm| wvd|