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

Understanding the Deltoid Phenomenon in the Perspective 3-Point Problem

“…But, by Lemma 5, upon making the substitution ( 8), (7) divided by d 4 2 is equal to (14) times η 2 . We see that the discriminant of (7) (as a polynomial in λ 1 and λ 3 ) equals the discriminant of ( 14) (as a polynomial in ρ 1 and ρ 2 ) times d 12 2 η 6 / (µ 11 µ 32 − µ 12 µ 31 ) 3 . This is ∆ times d 12 2 η 6 / (µ 11 µ 32 − µ 12 µ 31 ) 3 .…”

“…An important quantity for analyzing the P3P Problem is the Gramian determinant associated with the unit vectors pointing from the camera's optical center to the control points. Following [12], [13] and [15], this will be denoted by η 2 . It is equal to…”

“…Unfortunately, the system typically produces multiple positive-valued solutions, as many as four of these. Understanding how many such solutions result for given values of the parameters a, b, c, α, β and γ has been the focus of much investigation over the past few decades (for example, [1], [2], [11], [12], [13], [15]). Similar to the situation for a single polynomial in a single variable, this question is closely related to determining which parameter values produce a repeated solution to the system, and this amounts to examining a quantity, the "discriminant," based on the parameter values.…”

“…As explained in [10], [12], [13], [15] and [16], Grunert's system has a repeated positive solution if and only if one of the possible positions for the camera's optical center is on the circular cylinder based on the circle containing the three control points. This is the so-called "danger cylinder."…”

“…Following the practice adopted in [11], [12], [13] and [15], a special rectangular coordinate system will be used, though this in no way restricts the nature of the problem. The three coordinates are called x, y and z.…”