Understanding the Deltoid Phenomenon in the Perspective 3-Point Problem

Abstract

<p>Concerning the Perspective 3-Point (P3P) Problem, Grunert's system of three quadratic equations has a repeated solution if and only if the cubic polynomial introduced by Finsterwalder has a repeated root. This polynomial is shown to be equivalent to a particularly simple cubic polynomial with complex coefficients that provides surprising geometric insight into the P3P problem. In particular, (1) its discriminant can be written using the formula for the standard deltoid, and (2) this discriminant vanishes on a surface that approaches a deltoid shape when the camera is moved infinitely far from the control points in a direction perpendicular to the control points plane (the ``limit case"). These two facts have been previously reported, but obscure reasoning was required to establish them. In contrast, the present article uses the newly discovered cubic polynomial to easily produce the first fact, which then provides a basis for better understanding the second fact. Geometric insight into the P3P solution points in the limit case is also provided.</p>