A tour d’horizon of de Casteljau’s work

Abstract

Whilst Paul de Casteljau is now famous for his fundamental algorithm of curve and surface approximation, little is known about his other findings. This article offers an insight into his results in geometry, algebra and number theory. Related to geometry, his classical algorithm is reviewed as an index reduction of a polar form. This idea is used to show de Casteljau's algebraic way of smoothing, which long went unnoticed. We will also see an analytic polar form and its use in finding the intersection of two curves. The article summarises unpublished material on metric geometry. It includes theoretical advances, e.g., the 14-point strophoid or a way to link Apollonian circles with confocal conics, and also practical applications such as a recurrence for conjugate mirrors in geometric optics. A view on regular polygons leads to an approximation of their diagonals by golden matrices, a generalisation of the golden ratio. Relevant algebraic findings include matrix quaternions (and anti-quaternions) and their link with Lorentz' equations. De Casteljau generalised the Euclidean algorithm and developed an automated method for approximating the roots of a class of polynomial equations. His contributions to number theory not only include aspects on the sum of four squares as in quaternions, but also a view on a particular sum of three cubes. After a review of a complete quadrilateral in a heptagon and its angles, the paper concludes with a summary of de Casteljau's key achievements. The article contains a comprehensive bibliography of de Casteljau's works, including previously unpublished material.

Keywords: de-Casteljau algorithm, Curves, Surfaces, Blossoming, Polar Form, Quasi-Interpolation, Smoothing, Focal Splines, Metric Geometry, Geometric Optics, Quaternions, Regular Polygons, Euclidean Algorithm, 14-point strophoid