平面曲线生成的几何优化问题
Geometric optimization problems generated by plane curves
AI总结：
该研究探讨平面内三条正则曲线构成的几何优化问题，证明距离极值对应的三条直线共点或平行，还指出切线约束的线段极值问题是其特例，并分析了退化情形。
AI中文摘要：
设γ₁和γ₂为平面内的正则C¹光滑曲线，γ为同一平面内的正则C²光滑曲线。考虑所有三点组(A, A₁, A₂)，其中A∈γ、A₁∈γ₁、A₂∈γ₂，满足A₁≠A₂且直线A₁A₂是γ在A处的法线。我们证明：若γ的曲率非零，且三点组(A⁰, A₁⁰, A₂⁰)是点A₁与A₂之间距离|A₁A₂|的局部极大值或局部极小值，则以下三条直线要么交于一点，要么互相平行：γ₁在A₁⁰处的法线、γ₂在A₂⁰处的法线，以及垂直于A₁⁰A₂⁰且过γ在A⁰处曲率中心的直线。当γ为给定中心O的圆时，该优化问题的特殊情形与已部分研究的问题一致，即寻找局部最短（或局部最长）非退化线段[A₁A₂]，满足A₁∈γ₁、A₂∈γ₂且O∈A₁A₂。我们还证明，看似不同的问题——寻找局部最短（或局部最长）非退化线段[A₁A₂]，满足A₁∈γ₁、A₂∈γ₂且直线A₁A₂与γ相切——本质上也是上述优化问题的特殊情形。我们详细考虑了该设定下自然出现的“退化情形”，例如γ₁或γ₂与γ重合，或最优直线A₁⁰A₂⁰与γ₁或γ₂中至少一条相切的情况。
英文摘要：
Let $γ_1$ and $ γ_2$ be regular $C^1$-smooth curves in the plane and $γ$ be a regular $C^2$-smooth curve in the same plane. Consider all triples of points $(A, A_1, A_2)$, $A\in γ$, $A_1\in γ_1$, $A_2\in γ_2$, such that $A_1\neq A_2$, and the line $A_1 A_2$ is the normal to $γ$ at $A$. We show that, if $γ$ has non-vanishing curvature and the triple $(A^0, A_1^0,A_2^0 )$ is a local maximum or a local minimum for the distance $|A_1A_2|$ between the points $A_1$ and $A_2$, then the following three lines either meet at a single point or are parallel: the normal to $γ_1$ at $A_1^0$, the normal to $γ_2$ at $A_2^0$ and the line which is perpendicular to $A_1^0A_2^0$, and passing through the center of curvature of $γ$ at $A^0$. The particular case of this optimization problem, when $γ$ is a circle with a given center $O$, coincides with the already partially studied problem of finding the locally shortest (or the locally longest) non-degenerate segments $[A_1A_2]$ such that $A_1\inγ_1$, $A_2\inγ_2$ and $O\in A_1A_2$. We also show that the seemingly different problem of finding the locally shortest (or the locally longest) non-degenerate segments $[A_1A_2]$, such that $A_1 \in γ_1$, $A_2 \in γ_2$, and the line $A_1A_2$ is tangent to $γ$ is also, in essence, a particular case of the above optimization problem. We consider in detail the ``degenerate cases'' naturally appearing in this setting (when, for instance, $γ_1$ or $γ_2$ coincide with $γ$, or when the optimal line $A_1^0A_2^0$ is tangent to at least one of $γ_1$ or $γ_2$).
