# On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs

## TL;DR

This paper introduces canonical parametrizations for confocal coordinate systems, constructs non-planar analogues of incircular nets on quadrics, and explores their connections with classical deformations and three-dimensional webs, providing explicit constructions and illustrations.

## Contribution

It presents new canonical parametrizations, constructs non-planar incircular nets on quadrics, and establishes their connections with classical deformations and 3D webs, including explicit formulas and illustrations.

## Key findings

- Existence of octahedral webs of Blaschke-type surfaces
- Construction of 3D extensions of incircular nets
- Introduction of conical octahedral grids in Minkowski space

## Abstract

Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics which are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines which are diagonally related to lines of curvature is proven theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.