Formula for Dupin Cyclidic Cube and Miquel Point

Dupin cyclides are surfaces conformally equivalent to a torus, a circular cone, or a cylinder. Their patches admit rational bilinear quaternionic Bézier parametrizations and are used in geometric design and architecture. Dupin cyclidic cubes are a natural trivariate generalization of Dupin cyclide patches. In this article, we derive explicit formulas for control points and weights of rational trilinear quaternionic Bézier parametrizations of Dupin cyclidic cubes and relate them with the classical construction of the Miquel point.