For a pair of points in a smooth closed convex planar curve $$\gamma $$ 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; mid-parallel tangent locus and affine evolute of $$\gamma $$ γ . In this paper, we generalized these concepts by considering the envelope of the intermediate lines. For a pair of points of $$\gamma $$ 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 $$\gamma $$ 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; the curve $$\gamma $$ γ 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.

中文翻译：

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平面曲线中间线的包络

对于光滑闭合凸平面曲线 $$\gamma $$ γ 中的一对点，它的中线是包含它的中点和相应切线对的交点的线。众所周知，中线的包络 (EML) 是由三个仿射不变量集的并集形成的：仿射包络对称集；$$\gamma $$ γ 的中平行切线轨迹和仿射渐近线。在本文中，我们通过考虑中间线的包络来概括这些概念。对于$$\gamma $$ γ 的一对点，它的中间线是包含中间点的线和对应的切线对的交点。这里，我们给出了曲线$$\gamma $$ γ 的中间线包络（ EIL ），并证明当中间点与中点不同时，该集合由三个不相连的集合形成：中间线的仿射包络；曲线 $$\gamma $$ γ 本身和中间平行的切线轨迹。当中间点与中点重合时，EIL 与 EML 重合，因此这些集合是连接的。此外，我们介绍了奇点理论的一些标准技术，并使用它们来解释该集合的局部行为。