Envelope of mid-planes of a surface and some classical notions of affine differential geometry

TL;DR

This paper explores the relationship between mid-planes of a convex surface, their limits, and classical affine differential geometry concepts, revealing new geometric connections and properties of the affine mid-planes evolute.

Contribution

It establishes a novel link between mid-plane envelopes and classical affine differential geometry notions, introducing the affine mid-planes evolute and analyzing its properties.

Findings

Limit of mid-planes is the Transon plane in a given direction.

Envelope of mid-planes is non-empty for at most six directions.

The affine mid-planes evolute is a regular surface under generic conditions.

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.