A canonical discrete analogue of the classical circular cross sections of ellipsoids and their isometric deformation

TL;DR

This paper introduces a novel discretization method to construct discrete analogues of ellipsoids composed of planar quadrilaterals, which preserve classical geometric properties and admit isometric deformations similar to continuous ellipsoids.

Contribution

It presents a new explicit construction of discrete ellipsoids using a discretization procedure, including their geometric properties and deformation behavior, extending classical continuous results.

Findings

Discrete ellipsoids are composed of planar quadrilaterals.

They form pairs of dual polyhedra with classical geometric properties.

Discrete circles in these ellipsoids lie in parallel planes and deform isometrically.

Abstract

Based on a novel discretization procedure which has recently been proposed and applied in the construction of a canonical discrete analogue of confocal coordinate systems, an explicit method of constructing discrete analogues of ellipsoids is recorded. These discrete ellipsoids are entirely composed of planar quadrilaterals and come in pairs of combinatorially dual polyhedra, which together form princpal binets, a discrete counterpart of curvature line parametrizations. They exhibit a variety of algebraic and geometric properties which are classical in the case of their continuous counterparts. In a special case, it is demonstrated that the diagonals of the quadrilaterals of these discrete ellipsoids form two families of closed planar polygons. These polygons may be regarded as the discrete analogues of the classical circular cross sections of an ellipsoid. As in the continuous case, the discrete circles of any family lie in parallel planes and degenerate at the ``boundary'' of the discrete ellipsoids to ``umbilic vertices'' or ``umbilic edges''. Moreover, in analogy with the continuous theory, discrete ellipsoids admit a one-parameter family of deformations that preserve these discrete circles.