Geometric Conditions for the Existence or Non-existence of a Solution to the Perspective 3-Point Problem

Abstract

Direct and fairly simple geometric criteria are proved to be necessary for the Perspective 3-Point (P3P) Problem to have a real solution point. This is so under the assumption that the three control points are at the vertices of an acute triangle. Collectively, these criteria appear to be sufficient as well, based on substantial experimental evidence. Proving the necessity of some of the criteria does not involve the acute triangle assumption, and so these are required for obtuse and right triangles as well. While motivated by the P3P Problem, the results are actually concerned with various constraints among six of the angles that occur in a tetrahedron. Therefore, the results likely have other applications.

Appendix A

The “algebraic proof” of Theorem 3 begins with an arbitrary tetrahedron $\Delta ABCP$. Consider the system of evident equations and inequalities involving the interior angles of all four faces the tetrahedron. There are twelve such angles, six of which are $\angle A = \angle CAB$, $\angle B = \angle ABC$, $\angle C = \angle BCA$, $\alpha = BPC$, $\beta = \angle CPA$ and $\gamma = \angle APB$. The other six will be denoted $\alpha ' = \angle PCB$, $\beta ' = \angle PAC$, $\gamma \,' = \angle PBA$, $\alpha '' = \angle CBP$, $\beta '' = \angle ACP$ and $\gamma \,'' = \angle BAP$. All the angles are between 0 and $\pi$. The sum of the three angles at a particular vertex cannot exceed $2\pi$. Of course, the sum of the three angles for a particular face must equal $\pi$.

To be more specific, we begin with this system of equations and inequalities: $\angle A + \angle B +\angle C = \pi$, $\alpha + \alpha ' + \alpha '' = \pi$, $\beta + \beta ' + \beta '' = \pi$, $\gamma + \gamma \,' + \gamma \,'' = \pi$, $\angle A > 0$, $\angle B > 0$, $\angle C > 0$, $\alpha > 0$, $\beta > 0$, $\gamma > 0$, $\alpha ' > 0$, $\beta ' > 0$, $\gamma \,' > 0$, $\alpha '' > 0$, $\beta '' > 0$, $\gamma \,'' > 0$, $\alpha + \beta + \gamma < 2\pi$, $\alpha < \beta + \gamma$, $\beta < \gamma + \alpha$, $\gamma < \alpha + \beta$, $\angle A + \beta ' + \gamma \,'' < 2\pi$, $\angle A < \beta ' + \gamma \,''$, $\beta ' < \gamma \,'' + \angle A$, $\gamma \,'' < \angle A + \beta '$, $\alpha '' + \angle B + \gamma \,' < 2\pi$, $\alpha '' < \angle B + \gamma \,'$, $\angle B < \gamma \,' + \alpha ''$, $\gamma \,' < \alpha '' + \angle B$, $\alpha ' + \beta '' + \angle C < 2\pi$, $\alpha ' < \beta '' + \angle C$, $\beta '' < \angle C + \alpha '$, $\angle C < \alpha ' + \beta ''$.

This system can easily be reduced to a system of strict inequalities involving eight unknowns, by substituting $\pi - \angle A - \angle B$ for $\angle C$, $\pi - \alpha - \alpha '$ for $\alpha ''$, $\pi - \beta - \beta '$ for $\beta ''$, and $\pi - \gamma - \gamma \,'$ for $\gamma \,''$. Using Fourier–Motzkin elimination [10], one can now eliminate the three primed variables, leaving a system of strict inequalities in $\angle A$, $\angle B$, $\alpha$, $\beta$ and $\gamma$. The result is a system consisting of inequalities that are trivial or sphere-based rules (expressed without using $\angle C$). The following Mathematica code⁷ proceeds along these lines to produce such a system. The only resulting non-trivial rules are essentially Items 1 and 2 in Conjecture 1; Item 3 follows from these, as indicated by Proposition 2.