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

Abstract

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 an unexpected connection between these classical notions of affine differential geometry and the apparently unrelated concept of envelope of mid-planes. 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.