Real circles tangent to 3 conics

TL;DR

This paper investigates the number of circles tangent to three conics, establishing a generic count of 184 complex solutions, providing explicit real examples, and developing computational tools including hill-climbing algorithms and machine learning models.

Contribution

It characterizes the number of tangent circles to three conics, provides explicit real examples, and introduces computational methods for predicting and finding such circles.

Findings

184 complex tangent circles generically exist for three conics

An explicit example with 136 real tangent circles is provided

A machine learning model predicts the number of tangent circles for given conics

Abstract

In this paper we study circles tangent to conics. We show there are generically $184$ complex circles tangent to three conics in the plane and we characterize the real discriminant of the corresponding polynomial system. We give an explicit example of $3$ conics with $136$ real circles tangent to them. We conjecture that 136 is the maximal number of real circles. Furthermore, we implement a hill-climbing algorithm to find instances of conics with many real circles, and we introduce a machine learning model that, given three real conics, predicts the number of circles tangent to these three conics.