The problem of checking the equivalence of two sets of polynomial curv
es (in a plane), each set of total degree n, under the affine transfor
mations of translation, rotation, scaling and mirror reflection is stu
died. An optimal THETA(n log n) time algorithm for this problem is pre
sented. This algorithm is based on efficient computation of Bezier pol
ygons by using some polynomial manipulations that employ FFT methods.