Affine geometry, curve flows, and invariant numerical approximations (Q1354667)

The authors give a detailed discussion of equiaffine geometry, which includes new geometric approaches to the equiaffine normal and curvature. They discuss finite difference approximations of Euclidean and affine differential invariants. Finally, they provide some new solutions to the affine curvature flow. The main results are the following theorems: Theorem 1. Let \(\Gamma\) be a short arc and let \(d(A,B)\) be the equiaffine distance between its endpoints \(A\) and \(B\), i.e., twice the cube root of the area of its support triangle \(APB\). Let \(M\) be the midpoint of the chord \(AB\). Then, a mean value for the equiaffine frame \((t,n)\), consisting of the tangent and normal to \(\Gamma\), is represented by \(t_{av}= {B-A\over d(A,B)}\), and \(n_{av}= 4{M-P\over d(A,B)^2}\). Theorem 2. Let \(\Gamma\) be a short arc of a smooth, convex curve, with endpoints \(A\), \(B\). Let \(d(A,B)\) denote the equiaffine distance from \(A\) to \(B\). Then, a mean value of the equiaffine curvature \(k\) over \(\Gamma\) is represented by \(\widetilde k_\Gamma= 8{1+ \rho(A,B)\over d(A, B)^2}\), where \(\rho(A,B)\) denotes a certain (negatively valued) cross ratio. Theorem 3. Let \(P_0\), \(P_1\), \(P_2\), \(P_3\), \(P_4\) be five successive points on the convex curve \(\Gamma\). Let \(k\) be the affine curvature of \(\Gamma\) at \(P_2\), and let \(\widetilde k\) denote the affine curvature of the conic section passing through the five points. Let \(L_i= \int^{P_i}_{P_2} ds\), \(i= 0,\dots, 4\), denote the signed affine arc length of the conic form \(P_2\) to \(P_i\); in particular, \(L_2= 0\). We assume that each \(L_i\) is small. Then, the following expansion is valid: \(\widetilde k= k+{1\over 5} (\sum^4_{i=0} L_i){dk\over dx}+{1\over 30} (\sum_{0\leq i\leq j\leq 4} L_iL_j){d^2k\over ds^2}+\cdots\)\ . The higher-order terms are cubic in the distances \(L_i\).