On the number of intersection points of lines and circles in $\mathbb R^3$

TL;DR

This paper investigates the maximum number of intersection points between lines and circles in three-dimensional space, establishing an upper bound under certain geometric constraints.

Contribution

It provides a new upper bound of O(n^{3/2}) for intersection points when no large subset of curves lies on a low-degree algebraic surface.

Findings

Maximum intersection points are O(n^{3/2}) under given conditions.

No large subset of curves on a low-degree surface implies this bound.

The result advances understanding of geometric incidences in 3D space.

Abstract

We consider the following question: Given $n$ lines and $n$ circles in $\mathbb{R}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families. We prove that if there are no $n^{1/2}$ curves (lines or circles) lying on an algebraic surface of degree at most two, then the number of these intersection points is $O(n^{3/2})$.