Envelope of intermediate lines of a plane curve
For a pair of points in a smooth closed convex planar curve $γ$, 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 $γ$. In this paper, we generalized these concepts by considering the envelope of the intermediate lines. For a pair of points of $γ$, 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 $γ$ 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 $γ$ 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.