Offset-rational sinusoidal spirals in Bézier form

Mineur et al. [Mineur, Y., Lichah, T., Castelain, J.M., Giaume, H., 1998. A shape controlled fitting method for Bézier curves. Computer Aided Geometric Design 15 (9), 879–891] presented a remarkable family of Bézier curves, characterized by a control polygon with constant angle at each vertex and constant ratio between lengths of consecutive edges. We note that such curves belong to a classical family of polynomial sinusoidal spirals that encompasses Tschirnhausen's cubic, and can also be expressed as p-Bézier curves (Bézier curves in polar coordinates). These spirals have rational offsets, obtained simply by adding two p-Bézier curves, namely the spiral and a circle. Such a construction includes the offset to a parabola.