Formula for Dupin Cyclidic Cube and Miquel Point

TL;DR

The paper develops a complete quaternionic Bézier framework for Dupin cyclidic cubes by deriving explicit control points and weights for the trivariate map $F(s,t,u)=U/W$, where $U,W$ are quaternion-valued polynomials, and by tying the construction to the Miquel point. It starts from inversions and the Study quadric to first realize a DC cube with $p_0=\infty$ and then uses Möbius inversions to obtain general finite corner data. Key contributions include closed-form formulas for the seven finite corner weights and three equivalent expressions for the eighth weight $w_7$, together with Miquel-point expressions for the eighth corner $p_7$. The resulting parametrization supports modeling of 3D cyclidic nets and provides a practical tool for geometric design and architecture.

Abstract

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.

Figures (4)

• Figure 1: A Dupin cyclide principal patch.
• Figure 2: Steps of the construction of a DC cube.
• Figure 3: DC cube construction steps with one control point on infinity.
• Figure 4: A 3D-cyclidic net.

Theorems & Definitions (15)

• Remark 3.1
• Lemma 3.2
• Example 3.3
• Remark 3.4
• Remark 3.5
• Lemma 3.6
• proof
• Theorem 4.1
• proof
• Corollary 4.2