Z-1 = Zo -1(1 - px - qy - !kxxx2 - !kyyy2 - kxyxy) + O(x3 ), (3b) with normalized curvatures defined by

Second-Order Flows and Deformations

be given exactly by second-order polynomials in the image coordinates. For planes, the second-order flow model is globally valid since all higherorder Taylor coefficients vanish. For curved patches, Equation 3b is approximate and so a second-order flow model is only locally valid in this case. However, surface curvatures already influence the flow at secondorder. This leads us to believe that the second-order flow is the most fundamental image flow field. It is exact for planes and locally valid for quadric patches. It also makes explicit the fact that curved patches should not be approximated by planar facets, unless the local radii of curvature are large compared to object distance.