|Sfb 288 Differential Geometry and Quantum Physics|
Discrete Hashimoto surfaces and a doubly discrete smoke ring flow.
Bäcklund transformations for smooth and "space discrete" Hashimoto surfaces are discussed and a geometric interpretation is given. It is shown that the complex curvature of a discrete space curve evolves with the discrete nonlinear Schrödinger equation (NLSE) of Ablowitz and Ladik, when the curve evolves with the Hashimoto or smoke ring flow. A doubly discrete Hashimoto flow is derived and it is shown, that in this case the complex curvature of the discrete curve obeys Ablovitz and Ladik's doubly discrete NLSE. Elastic curves (curves that evolve by rigid motion only under the Hashimoto flow) in the discrete and doubly discrete case are shown to be the same.
Conformally Symmetric Circle Packings. A generalization of Doyle spirals
A. I. Bobenko and T. HoffmannAbtract:
From the geometric study of the elementary cell of hexagonal circle packings --- a flower of 7 circles --- the class of conformally symmetric circle packings is defined. Up to Moebius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are given explicitly. It is shown, that these circle packings can be viewed as discretizations of the quotient of two Airy functions.
This paper contains JAVA applets that let you experiment with the circle packings directly.