An Elliptic Curve Based Solution to the Perspective-Three-Point Problem

TL;DR

This paper introduces an elliptic curve-based approach to solving the Perspective-Three-Point problem, offering a more accurate method that reveals a novel connection between P3P and elliptic curves, with potential for future advancements.

Contribution

It presents a new P3P solver based on elliptic curves and uncovers a fundamental link between P3P and cryptographic elliptic curves, expanding the theoretical understanding.

Findings

The new method is more accurate than Lambda Twist under certain conditions.

Both methods depend on computing a single root of a cubic polynomial.

The elliptic curve connection opens avenues for further research.

Abstract

The Perspective-Three-Point Problem (P3P) is solved by first focusing on determining the directions of the lines through pairs of control points, relative to the camera, rather than the distances from the camera to the control points. The analysis of this produces an efficient, accurate and reasonably simple P3P solver, which is compared with a state-of-the-art P3P solver, "Lambda Twist." Both methods depend on the accurate computation of a single root of a cubic polynomial. They have been implemented and tested for a wide range of control-point triangles, and under certain reasonable restrictions, the new method is noticably more accurate than Lambda Twist, though it is slower. However, the principal value of the present work is not in introducing yet another P3P solver, but lies rather in the discovery of an intimate connection between the P3P problem and a special family of elliptic curves that includes curves utilized in cryptography. This holds the potential for further advances in a number of directions. To make this connection, an interesting spherical analogue of an ancient "sliding" problem is stated and solved.