Let γ be as above, and fix t. We want to find the radius ρ of a parametrized circle which matches γ in its zeroth, first, and second derivatives at t. Clearly the radius will not depend on the position γ (t), only on the velocity γ ′(t) and acceleration γ ″(t). There are only three independent scalars that can be obtained from two vectors v and w, namely v · v, v · w, and w · w. Thus the radius of curvature must be a function of the three scalars | γ ′(t) |, | γ ″(t) | and γ ′(t) · γ ″(t).