108 research outputs found
Orthogonal nets and Clifford algebras
A Clifford algebra model for M"obius geometry is presented. The notion of
Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced,
and the structure equations for adapted frames are derived. These equations are
discretized and the geometry of the occuring discrete nets and sphere
congruences is discussed in a conformal setting. This way, the notions of
``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal
systems'' are obtained --- the latter as a generalization of discrete
orthogonal systems in Euclidean space. The relation of a Cauchy problem for
discrete orthogonal nets and a permutability theorem for the Ribaucour
transformation of smooth orthogonal systems is discussed.Comment: Plain TeX, 16 pages, 4 picture
Incircular nets and confocal conics
We consider congruences of straight lines in a plane with the combinatorics
of the square grid, with all elementary quadrilaterals possessing an incircle.
It is shown that all the vertices of such nets (we call them incircular or
IC-nets) lie on confocal conics.
Our main new results are on checkerboard IC-nets in the plane. These are
congruences of straight lines in the plane with the combinatorics of the square
grid, combinatorially colored as a checkerboard, such that all black coordinate
quadrilaterals possess inscribed circles. We show how this larger class of
IC-nets appears quite naturally in Laguerre geometry of oriented planes and
spheres, and leads to new remarkable incidence theorems. Most of our results
are valid in hyperbolic and spherical geometries as well. We present also
generalizations in spaces of higher dimension, called checkerboard IS-nets. The
construction of these nets is based on a new 9 inspheres incidence theorem.Comment: 33 pages, 24 Figure
Discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. A discrete Lawson correspondence
The main result of this paper is a discrete Lawson correspondence between
discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a
correspondence between two discrete isothermic surfaces. We show that this
correspondence is an isometry in the following sense: it preserves the metric
coefficients introduced previously by Bobenko and Suris for isothermic nets.
Exactly as in the smooth case, this is a correspondence between nets with the
same Lax matrices, and the immersion formulas also coincide with the smooth
case.Comment: 13 page
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