Revisiting Burmester theory with complex forms of Bottema's instantaneous invariants

The objective of this study is to take advantage of using the concept of complex numbers for instantaneous geometric properties of planar motion of rigid bodies. We consider that both the fixed and the moving planes are Gaussian. Then we give Bottema's instantaneous invariants in complex forms and we study the kinematic geometry of infinitesimally separated positions of a moving Gaussian plane regarding this formulation. The followed analytical method provides a straightforward way to describe order properties of motion. We obtain the complex forms of the inflection circle, cubic of stationary curvature and cubic of twice stationary curvature. Moreover, we give the existence conditions of Ball, Burmester and Ball-Burmester points