Envelope of intermediate lines of a plane curve

TL;DR

This paper extends the concept of the envelope of mid-lines of a convex curve to intermediate lines, characterizing its structure and singularities using singularity theory techniques.

Contribution

It introduces the envelope of intermediate lines for convex curves, describing its structure and singularities, generalizing previous affine invariants related to mid-lines.

Findings

The envelope of intermediate lines consists of three disconnected sets when the intermediate point differs from the mid-point.

When the intermediate point is the mid-point, the envelope coincides with the envelope of mid-lines, forming a connected set.

Singularity theory techniques are used to analyze the local behavior of the envelope of intermediate lines.

Abstract

For a pair of points in a smooth closed convex planar curve $\gamma$, its mid-line is the line containing its mid-point and the intersection point of the corresponding pair of tangent lines. It is well known that the envelope of the mid-lines ($EML$) is formed by the union of three affine invariants sets: Affine Envelope Symmetry Sets ($AESS$); Mid-Parallel Tangent Locus ($MPTL$) and Affine Evolute of $\gamma$. In this paper, we generalized these concepts by considering the envelope of the intermediate lines. For a pair of points of $\gamma$, its intermediate line is the line containing an intermediate point and the intersection point of the corresponding pair of tangent lines. Here, we present the envelope of intermediate lines ($EIL$) of the curve $\gamma$ and prove that this set is formed by three disconnected sets when the intermediate point is different from the mid-point: Affine Envelope of Intermediate Lines ($AEIL$); the curve $\gamma$ itself and the Intermediate-Parallel Tangent Locus ($IPTL$). When the intermediate point coincides with the mid-point, the $EIL$ coincides with the $EML$, and thus these sets are connected. Moreover, we introduce some standard techniques of singularity theory and use them to explain the local behavior of this set.