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 here shown to be obtainable from a particularly simple
cubic polynomial with complex coefficients via a simple Möbius
transformation. This provides surprising geometric insight into the P3P
problem. In particular, (1) the discriminant of Finsterwalder’s
polynomial can be written using the formula for the standard deltoid
curve, 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. A detailed geometric description of the
P3P solution points in the “limit case” is also provided.