A minimal closed-form solution to the conic based on self-polar triangle

TL;DR

This paper introduces a novel, minimal closed-form solution for conics using self-polar triangles, enabling straightforward derivation of conditions for real, non-degenerate conics from minimal point-line configurations.

Contribution

It presents a new method leveraging self-polar triangles to derive closed-form solutions for conics from minimal configurations, with criteria for real, non-degenerate conics.

Findings

Closed-form solutions for conics from minimal configurations.

Criteria for real, non-degenerate conics.

Validation through illustrative examples.

Abstract

In this paper, we use the properties of the self-polar triangle to not only show a novel method for a basic point-line enumerative problem of conics, but also present a series of closed-form solutions to the conics from all minimal configurations of points and lines in general position. These closed-form formulae may allow us to derive easily the algebraic and geometric conditions which characterize when the obtained conic is real and non-degenerate, so we propose a criterion for a non-degenerate real conic from each of all minimal configurations. The correctness of our results is validated by some examples.