Affine Invariants of Certain Pairs of Curves and Surfaces

For two curves in a plane or two surfaces in ordinary space various projective invariants have been given by different authors. Obviously each projectiva invariant is also an affine invariant, that is, an invariant with respect to the group of affine transformations. However in certain cases there are affine invariants which are not projective invariants. The purpose of the present paper is to study these cases giving affine invariants, as well as their affine and metrical characterization, for the following cases: (a) two curves in a plane having a common tangent at two ordinary points (b) two curves in a plane intersecting at an ordinary point (c) two surfaces in ordinary space having a common tangent plane at two ordinaiy points (d) two surfaces in ordinary space having a common tangent line but distinct tangent plane at two ordinary points. For the cases (a),(b) of plane curves we shall consider the neighborhoods of the second and the third order of the curves at the considered points. For the cases (c), (d) of two surfaces in ordinary space we shall consider only the neighborhoods of the second order of the surfaces at the considered point