Envelope of Mid-Planes of a Surface and Some Classical Notions of Affine Differential Geometry

For a pair of points in a smooth locally convex surface in 3-space, its mid-plane is the plane containing its mid-point and the intersection line of the corresponding pair of tangent planes. In this paper we show that the limit of mid-planes when one point tends to the other along a direction is the Transon plane of the direction. Moreover, the limit of the envelope of mid-planes is non-empty for at most six directions, and, in this case, it coincides with the center of the Moutard’s quadric. These results establish a connection between these classical notions of affine differential geometry and the apparently unrelated concept of envelope of mid-planes of a surface. We call the limit of envelope of mid-planes the affine mid-planes evolute and prove that, under some generic conditions, it is a regular surface in 3-space.