Understanding the Deltoid Phenomenon in the Perspective 3-Point Problem

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.