Canal surfaces as Bézier curves using mass points
Résumé
The paper aims to connect the Bézier curves domain to another known as the Minkowski–Lorentz space for CAGD purposes. The paper details these connections. It provides new algorithms for surface representations and surfaces joining. Some $G^1$-blends between canal surfaces illustrate the results with a seahorse sketched. It is well known that rational quadratic Bézier curves define conics. Here, the use of mass points offers the definition of a semi-conic or a branch of hyperbola in the Euclidean plane. Moreover, the choice of an adequate non-degenerate indefinite quadratic form makes a non-degenerate central conic seen as a unit circle. That is not possible in the homogeneous coordinates background. The rational quadratic Bézier curves using mass points provide the modelling of canal surfaces with singular points. These are embedded in the Minkowski–Lorentz space. In that space, these curves are circular arcs which look like ellipse or hyperbola arcs. In the Minkowski–Lorentz space, oriented spheres and oriented planes of $\mathbb{R}^3$ are points on the unit sphere, the points of $\mathbb{R}^3$ and the point at infinity are vectors laying on the light-cone. A particular case of canal surfaces is the cyclides of Dupin. In the Minkowski–Lorentz space, the modelling of any Dupin cyclides patch is completed by a family of algorithms. Each family depends only on the number of singularities known for the Dupin cyclide part. The paper ends with an example of a $G^1$ connection between two Dupin cyclides. All previous results are finally applied in a seahorse shape design.