A Three-Dimensional Laguerre Geometry and Its Visualization

The aim of the present paper is to discuss in some detail the Laguerre geometry (cf. [1], [6]) which arises from the 3-dimensional real algebra L := R(ε), where ε = 0. This algebra generalizes the algebra of real dual numbers D = R(ε), where ε = 0. The Laguerre geometry over D is the geometry on the so-called Blaschke cylinder (Figure 1); the non-degenerate conics on this cylinder are called chains (or cycles, circles). If one generator of the cylinder is removed then the remaining points of the cylinder are in one-one correspondence (via a stereographic projection) with the points of the plane of dual numbers, which is an isotropic plane; the chains go over to circles and non-isotropic lines. So the point space of the chain geometry over the real dual numbers can be considered as an affine plane with an extra “improper line”. The Laguerre geometry based on L has as point set the projective line P(L) over L. It can be seen as the real affine 3-space on L together with an “improper affine plane”. There is a point model for this geometry, like the Blaschke cylinder, but it is more complicated, and belongs to a 7-dimensional projective space ([6, p. 812]). We are not going to use it. Instead, we describe P(L) as an extension of the affine space on L by “improper points” which will be described via lines, parabolas, and cubic parabolas. R Rε