
Journal for Geometry and Graphics 08 (2004), No. 2, 201213 Copyright Heldermann Verlag 2004 On Wallace Loci from the Projective Point of View Rolf Riesinger Patrizigasse 7/14, 1210 Vienna, Austria rolf.riesinger@chello.at [Abstractpdf] Let $\pi_k$ be the projection of an ndimensional projective space $\Sigma$ ($2\leq\,n<\infty$) from the point $B_k$ onto the hyperplane $\alpha_k$, $k = 1,\,\ldots, n+1$, and assume that $\alpha_1,...,\alpha_{n+1}$ are linearly independent. By the Wallace locus of $\pi_1,...,\pi_{n+1}$ we mean the set of all points X of $\Sigma$ whose images $\pi_1(X),...,\pi_{n+1}(X)$ are linearly dependent. In a Pappian nspace each Wallace locus is either the entire space or an algebraic hypervariety whose degree is at most n+1. In a Pappian plane a triangle ${B_1,B_2,B_3}$ and a trilateral ${\alpha_1,\alpha_2,\alpha_3}$ determine the same Wallace locus as the triangle ${\alpha_2\cap\alpha_3,\alpha_3\cap\alpha_1,\alpha_1\cap\alpha_3}$ and the trilateral ${B_2\vee\,B_3,B_3\vee\,B_1,B_1\vee\,B_2}$. An analogous exchange rule for $3\leq n < infty$ is not valid. For Wallace loci of a Pappian plane with collinear centers $B_1,B_2,B_3$ we exhibit a theorem wherefrom we get the Wallace theorems for all degenerate CayleyKlein planes by specialization. Thus we get the orthogonal and oblique Euclidean Wallace lines, the orthogonal and oblique pseudoEuclidean Wallace lines, and the isotropic Wallace lines and, by duality, the Wallace points of the dualEuclidean plane, of the dualpseudoEuclidean plane, and of the isotropic plane. Keywords: Triangle geometry, Wallace line, pedal line, Simson line, Wallace subspace. MSC: 51N15; 51M05 [ Fulltextpdf (224 KB)] for subscribers only. 