Measures and geometric probabilities for ellipses intersecting circles

TL;DR

This paper generalizes Santaló's measures and probabilities for a line segment intersecting a circle to the case of ellipses, deriving new geometric measures and hitting probabilities for lattice arrangements.

Contribution

It introduces generalized measures and probabilities for ellipses intersecting circles, extending prior line segment results to more complex shapes.

Findings

Derived measures for ellipses inside, enclosing, and intersecting circles.

Calculated hitting probabilities for lattice arrangements of circles.

Showed that line segment results are special cases of ellipse results.

Abstract

Santal\'o calculated the measures for all positions of a moving line segment in which it lies inside a fixed circle and intersects this circle in one or two points. From these measures he concluded hitting probabilities for a line segment thrown randomly onto an unbounded lattice of circles. In the present paper these results are generalized to ellipses instead of line segments. The respective measures for all positions of a moving ellipse in which it lies completely inside a fixed circle, encloses it, and intersects it in two or four points are derived. Then the hitting probabilities for lattices of circles are deduced. It is shown that the results for a line segment follow as special cases from those of the ellipse.