Envelope of Intermediate Lines of a Plane Curve

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; mid-parallel tangent locus 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; the curve $\gamma$ itself and the intermediate-parallel tangent locus. 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.