Understanding the Deltoid Phenomenon in the Perspective 3-Point (P3P) Problem

Abstract 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 Mobius 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, when regarded as a function of camera position, 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. Also presented are quartic polynomials whose real roots are the P3P solution point coordinates. A detailed geometric description of the P3P solution points in the ''limit case" is also supplied.