# 3264 Conics in a Second

## TL;DR

This paper demonstrates how enumerative and numerical algebraic geometry complement each other by computing the 3264 conics tangent to five given conics, using HomotopyContinuation.jl for efficient solutions.

## Contribution

It introduces a web interface leveraging HomotopyContinuation.jl to compute all solutions to a classical enumerative geometry problem efficiently.

## Key findings

- Successfully computed all 3264 solutions for a specific instance.
- Identified an example with all solutions being real.
- Showcased the integration of theoretical and numerical methods.

## Abstract

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. This article illustrates how these two fields complement each other. Our focus lies on the 3264 conics that are tangent to five given conics in the plane. We present a web interface for computing them. It uses the software HomotopyContinuation.jl, which makes this process fast and reliable. We discuss an instance where all 3264 solutions are real.