title: p-Bezier curves, spirals, and sectrix curves
abstract: We elucidate the connection between Bezier curves in polar coordinates, also called p-Bezier or focal Bezier curves, and certain families of spirals and sectrix curves. p-Bezier curves are the analogue in polar coordinates of nonparametric Bezier curves in Cartesian coordinates. Such curves form a subset of rational Bezier curves characterized by control points on radial directions regularly spaced with respect to the polar angle, and weights equal to the inverse of the polar radius. We show that this subset encompasses several classical sectrix curves, which solve geometrically the problem of dividing an angle into equal spans, and also spirals defining the trajectories of particles in central fields. First, we identify as p-Bezier curves a family of sinusoidal spirals that includes Tschirnhausen's cubic. Second, the trisectrix of Maclaurin and their generalizations, called arachnidas. Finally, a special class of epi spirals that encompasses the trisectrix of Delanges
keyphrases: p-Bezier curves spirals sectrix curves polar coordinates focal Bezier curves rational Bezier curves control points radial directions polar angle equal spans central fields sinusoidal spirals cubic trisectrix arachnidas epi spirals geometry angle division particle trajectories