On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs
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On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs Arseniy V. Akopyan1 · Alexander I. Bobenko2 · Wolfgang K. Schief3 Jan Techter2
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Received: 23 August 2019 / Revised: 5 March 2020 / Accepted: 28 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 that are isometric along asymptotic lines and circular crosssections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved 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. Keywords Confocal quadrics · Orthogonal coordinate system · 3-Web · Isometric deformation Mathematics Subject Classification 51B15 · 51N20 · 52Cxx · 53A60
1 Introduction Conics and quadrics have been the subject of extensive studies in both mathematics and physics since their discovery by the Ancient Greek [19,22]. The fascination
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123
Discrete & Computational Geometry
with confocal quadrics [21] over the centuries culminated in the famous Ivory and Graves–Chasles theorems which were given a modern perspective by Arnold [2]. Their gravitational properties were studied by Newton and Ivory (see, e.g., [2,17]) and deep connections with dynamical systems and spectral theory have also been established [27]. Confocal quadrics go hand in hand with confocal coordinate systems which have been employed, for instance, in the theory of separable linear differential equations such as the Laplace equation, which, in turn, is closely related to the theory of special functions such as (generalised) Lamé functions [16]. The present paper is based on the following, apparently novel, property of planar confocal quadrics. A confocal system of conics on the plane is represented by y2 x2 + = 1, λ+a λ+b
a > b,
(1)
where the parameter λ labels the individual conics. In [6], the privileged parametrisation √ a−b x sn(s, k) ns(t, k) , (2) = a−c , k= y cn(s, k) ds(t, k) a−c of
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