Generalized Burmester points computation by means of Bottema's instantaneous invariants and intrinsic geometry

This paper reports the algebraic derivations of the quintic polynomial equation whose solution gives the coordinates of generalized Burmester points (GBPs). Denoted with λ1 and λ2 the ratios of the first and second rate of change of curvature to curvature, respectively, the paths traced by GBPs have prescribed values of such ratios. When λ1=λ2=0, GBPs reduce to the (four) classical Burmester points. Our derivations, based on the properties of Cesàro's intrinsic geometry, led to a concise algebraic form of such polynomial coefficients. This availability allows expanding the field of application of Bottema's instantaneous invariants in higher-order mechanical approximation of any algebraic or parametric curve.