Real circles tangent to 3 conics

TL;DR

The paper addresses the problem of counting circles tangent to three conics, establishing that generically there are $184$ complex tritangent circles and demonstrating a real instance with $136$ real tritangent circles, leading to the conjecture that $136$ is the maximum. It derives the complex bound via intersection theory on the space of complete conics and a Veronese blowup, and analyzes the real discriminant to understand where the real count can change. Computational methods are then developed: hill-climbing to construct conic configurations with many real tangencies, and a data-driven approach using a neural-network-like predictor trained on hill-climb and random data to estimate counts from conic coefficients. The combination of algebraic geometry, numerical continuation with certification, and predictive modeling provides both explicit constructions (e.g., a $136$-real-circle instance) and practical tools to explore the real solution landscape, with data and code made available on MathRepo. These results connect classical tangency problems to modern computational techniques and offer a probabilistic lens on discriminant structure and real solubility.

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.

Figures (4)

• Figure 1: A red circle tangent to one blue hyperbola and two blue ellipses.
• Figure 2: A red circle tangent to three blue lines that form a triangle. The proof of \ref{['thm:136_R']} considers such a triangle and finds a nearby arrangement of three hyperbolas, such that there are 136 real circles tangent to the three hyperbolas. In fact, the arrangement of hyperbolas in the proof of \ref{['thm:136_R_intro']} is near the triangle shown in the picture.
• Figure 3: We show the first 50 conic arrangements that were used to prove \ref{['thm:all are possible']}. The number below each plot indicates the number of real tritangent circles. Starting from 92 real circles the arrangement all look similar to a triangle as in \ref{['fig2']}.
• Figure 4: Two histograms showing the distribution of the number of real circles tangent to a given configuration of conics. The top histogram shows the distribution on the data $\mathcal{D}_1$ we generated using the hill climbing algorithm and the bottom histogram shows the distribution on random data $\mathcal{D}_2$.

Theorems & Definitions (10)

• Theorem 1.3
• Conjecture 1.4
• Proposition 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3: The real discriminant
• proof
• proof : Proof of Theorem \ref{['thm:136_R_intro']}.
• Theorem 3.1