
TL;DR
This paper explores the lesser-known mathematical contributions of Paul de Casteljau across geometry, algebra, and number theory, revealing new insights and practical applications beyond his well-known curve algorithms.
Contribution
It uncovers previously unpublished results of de Casteljau in various mathematical fields, expanding understanding of his work beyond curve approximation.
Findings
De Casteljau's algebraic smoothing method
Analytic polar form for curve intersection
Approximation of polygon diagonals using golden matrices
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 The article summarises unpublished material on metric geometry.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…
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