formally speaking, the curves must be differentiable curves in the euclidean plane. the "fixed curve" is kept invariant; the "rolling curve" is subjected to a continuous congruence transformation such that at all times the curves are tangent at a point of contact that moves with the same speed when taken along either curve (another way to express this constraint is that the point of contact of the two curves is the instant centre of rotation of the congruence transformation). the resulting roulette is formed by the locus of the generator subjected to the same set of congruence transformations.