My old calculus book gives a formula for the curvature of a parametric arc in the plane — that is, an arc defined by two functions (x(t),y(t)) of one variable. For thirty years I didn’t think about the derivation of that formula. Just now it hit me (and I did the algebra to confirm) that, in terms of the complex plane (z=x+iy), the curvature formula is equivalent to
Im(z″/z′) / |z′|
This should improve my cubic approximations to transcendental curves.