A Projection Relationship Between the Centres of Curvature in Planar and Precessional Motion for Rolling Circles of Equal Radius

Abstract

The planar rolling of a moving circle on a fixed circle of the same radius and the corresponding precessional, i.e. spherical, motion are considered. Two previous papers addressed a special graphical construction of the centre of curvature for a point in the plane of symmetry of regular precession and a general property of the centre of curvature of an arbitrary point in spherical motion. The present paper investigates the relationship between the centre of curvature K0 of the corresponding planar trajectory and the centre of curvature K of the spatial trajectory. It is proved that, for circles of equal radius and an arbitrary regular point of the moving plane for which both centres are finite, the points O, K0, and K are collinear, where O is the fixed point of the spherical motion. As a practical consequence, the spatial centre of curvature is obtained by orthogonally projecting the current position of point P onto the line OK0.