On the Discriminant of Grunert’s System of Algebraic Equations and Related Topics

Abstract

Grunert’s system of equations is commonly used as a basis for mathematical investigations into the Perspective 3-Point Pose Problem, for camera resectioning and tracking. This consists of three quadratic equations involving three unknown distances. The discriminant of this system helps to determine the number of real-valued solutions, in terms of the system’s parameters. In its raw form, this is a very complicated and seemingly unintelligible polynomial. However, through a series of algebraic manipulations, this article manages to bring this polynomial into a far more sensible form. In addition, by making substitutions suggested by the system of equations, the discriminant is realized as a rational function of the Cartesian coordinates of the ambient space containing the control points. Moving perpendicular to the plane containing the control points, and moving away from this plane, cross sections of the surface on which this rational function vanishes approach a deltoid curve, together with the deltoid’s inscribed circle (a cross section of the danger cylinder). As long as such a cross section of the surface is not too close to the control points plane, it is homeomorphic to a union of the deltoid and its inscribed circle. The orthogonal projection of the deltoid onto the control points plane contains in its interior, the control points triangle’s orthocenter.