p-Bézier curves, spirals, and sectrix curves Original Research Article

We elucidate the connection between Bézier curves in polar coordinates, also called p-Bézier or focal Bézier curves, and certain families of spirals and sectrix curves. p-Bézier curves are the analogue in polar coordinates of nonparametric Bézier curves in Cartesian coordinates. Such curves form a subset of rational Bézier 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-Bézier curves a family of sinusoidal spirals that includes Tschirnhausenʹs cubic. Second, the trisectrix of Maclaurin and their generalizations, called arachnidas. Finally, a special class of epi spirals that encompasses the trisectrix of Delanges.