Geometric optimization problems generated by plane curves

Abstract

Let ${\gamma }_1$ and ${\gamma }_2$ be regular $C^1$-smooth curves in the plane and $\gamma$ be a regular $C^2$-smooth curve in the same plane. Consider all triples of points $(A,A_1,A_2)$, $A\in \gamma$, $A_1\in {\gamma }_1$, $A_2\in {\gamma }_2$, such that $A_1\ne A_2$, and the line $A_1{A}_2$ is the normal to $\gamma$ at $A$. We show that, if $\gamma$ 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 $|A_1{A}_2|$ 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 ${\gamma }_1$ at $A_1^0$, the normal to ${\gamma }_2$ at $A_2^0$ and the line which is perpendicular to $A_1^0{A}_2^0$, and passing through the center of curvature of $\gamma$ at $A^0$. The particular case of this optimization problem, when $\gamma$ 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 $[A_1{A}_2]$ such that $A_1\in {\gamma }_1$, $A_2\in {\gamma }_2$ and $O\in A_1{A}_2$. We also show that the seemingly different problem of finding the locally shortest (or the locally longest) non-degenerate segments $[A_1{A}_2]$, such that $A_1\in {\gamma }_1$, $A_2\in {\gamma }_2$, and the line $A_1{A}_2$ is tangent to $\gamma$ 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, ${\gamma }_1$ or ${\gamma }_2$ coincide with $\gamma$, or when the optimal line $A_1^0{A}_2^0$ is tangent to at least one of ${\gamma }_1$ or ${\gamma }_2$).