Curvature radius of conic sections: A kinematic derivation

Conics are relevant in many undergraduate course on classical physics, from free fall to Rutherford scattering. A pedagogical derivation of the intrinsic expressions of the curvature radius of different types of conic sections is presented. Our proof is carried out without resorting to any coordinate systems, but rather on using only elementary kinematic concepts together with basics of vector calculus and the very definition of conics. As a byproduct application of the present analysis, a simple and compact deduction of the Newton 'inverse square law' for gravitation from the three Kepler laws is also presented.