Bobillier Formula for One Parameter Motions in the Complex Plane

Abstract

This is a brief note expanding on the aspect of Fayet (2002, “Bobillier Formula as a Fundamental Law in Planar Motion,” Z. Angew. Math. Mech., 82(3), pp. 207–210), which investigates the Bobillier formula by considering the properties up to the second order planar motion. In this note, the complex number forms of the Euler Savary formula for the radius of curvature of the trajectory of a point in the moving complex plane during one parameter planar motion are taken into consideration and using the geometrical interpretation of the Euler Savary formula, Bobillier formula is established for one parameter planar motions in the complex plane. Moreover, a direct way is chosen to obtain Bobillier formula without using the Euler Savary formula in the complex plane. As a consequence, the Euler Savary given in the complex plane will appear as a particular case of Bobillier formula.