On an integrable multi-component Camassa–Holm system arising from Möbius geometry

In this paper, we mainly study the geometric background, integrability and peaked solutions of a ( 1 + n ) -component Camassa–Holm (CH) system and some related multi-component integrable systems. Firstly, we show this system arises from the invariant curve flows in the Möbius geometry and serves as the dual integrable counterpart of a geometrical ( 1 + n ) -component Korteweg–de Vries system in the sense of tri-Hamiltonian duality. Moreover, we obtain an integrable two-component modified CH system using a generalized Miura transformation. Finally, we provide a necessary condition, under which the dual integrable systems can inherit the Bäcklund correspondence from the original ones.

Generalised cyclidic nets for shape modelling in architecture

The aim of this article is to introduce a bottom-up methodology for the modelling of free-form shapes in architecture that meet fabrication constraints. To this day, two frameworks are commonly used for surface modelling in architecture: non-uniform rational basis spline modelling and mesh-based approaches. The authors propose an alternative framework called generalised cyclidic nets that automatically yield optimal geometrical properties for the envelope and the structural layout, like the covering with planar quadrilaterals or hexagons. This framework uses a base circular mesh and Dupin cyclides, which are natural objects of the geometry of circles in space, also known as Möbius geometry. This article illustrates how complex curved shapes can be generated from generalised cyclidic nets. It addresses the extension of cyclidic nets to arbitrary topologies with the implementation of a ‘hole-filling’ strategy and also demonstrates that this framework gives a simple method to generate corrugated shells.