For calibrated two-view geometry,
$$ x_2^\top E x_1=0,\qquad E=[t]_\times R. $$
A generic rotation matrix $R\in SO(3)$ has nine entries constrained by
$$ R^\top R=I,\qquad \det R=1. $$
The symmetric equation $R^\top R=I$ imposes six independent constraints—three unit-length and three orthogonality constraints—leaving
$$ 9-6=3\ \text{rotation DoF}. $$
Together with two translation-direction DoF, a generic relative pose therefore has
$$ 3_{\text{rotation}}+2_{\text{translation direction}}=5\ \text{DoF}, $$
so five point correspondences are minimal.
If the rotation axis is known, the rotational freedom reduces to only the angle:
$$ 1_{\text{angle}}+2_{\text{translation direction}}=3\ \text{DoF}. $$
Hence three point correspondences are minimal.
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Research focus. Existing methods already solve this three-point problem. The goal here is to understand the geometry behind the four solutions, connect the existing formulations, distinguish chart singularities from genuine data degeneracy, and explore globally compatible three-view extensions.
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| Formulation | Core equation | What it establishes |
|---|---|---|
| Generic essential variety | $2EE^\top E-\operatorname{tr}(EE^\top)E=0$ | Dimension 5, degree 10; basis of the calibrated five-point problem. |
| Fraundorfer et al. | $E=\begin{bmatrix}a&-b&c\\b&a&d\\e&f&0\end{bmatrix}$ | Three-point structured-$E$ solver; the essential cubics reduce to a quartic. |
| Sweeney et al. | $(q^2M+qC+K)t=0$ | Direct pose solver using the cotangent half-angle and a quadratic eigenvalue problem (QEP). |
| Guan et al. | One affine match gives three linear constraints. | Replaces three point matches with one affine correspondence; the known-vertical problem remains quartic. |
| Choi and Kim | $(\theta,\phi)$: yaw and planar translation direction | Planar motion has two observable DoF, giving a two-point minimal problem. |
Planar-motion two-point methods impose the stronger condition that translation also lies in a plane. They are related but do not describe the same three-DoF family, so they are background rather than the central comparison.
The accompanying constrained-E repository provides a controlled comparison of:
| Method | Implementation compared | Role |
|---|---|---|
| Sweeney QEP | Cotangent-half-angle QEP, $q=\cot(\theta/2)$ | Standalone baseline |
| Fraundorfer quartic | Structured-$E$ nullspace and quartic elimination | Standalone baseline |
| Chart-aware wrapper | Tangent QEP $\rightarrow$ cotangent QEP $\rightarrow$ structured-$E$ fallback | Wrapper around equivalent formulations, not a new solver family |