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Journal for Geometry and Graphics 08 (2004), No. 2, 201--213
Copyright Heldermann Verlag 2004

On Wallace Loci from the Projective Point of View

Rolf Riesinger
Patrizigasse 7/14, 1210 Vienna, Austria


Let $\pi_k$ be the projection of an n-dimensional 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 n-space 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 Cayley-Klein planes by specialization. Thus we get the orthogonal and oblique Euclidean Wallace lines, the orthogonal and oblique pseudo-Euclidean Wallace lines, and the isotropic Wallace lines and, by duality, the Wallace points of the dual-Euclidean plane, of the dual-pseudo-Euclidean plane, and of the isotropic plane.

Keywords: Triangle geometry, Wallace line, pedal line, Simson line, Wallace subspace.

MSC: 51N15; 51M05

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