A Closed-form Solution to the Wahba Problem for Pairwise Similar Quaternions

We present a closed-form solution to Wahba's problem in the quaternion domain for the special case of two vector observations. Existing approaches, including Davenport's $q$-method, QUEST, Horn's method, and ESOQ algorithms, recover the optimal quaternion through the eigendecomposition of a $4\times4$ matrix or iterative numerical methods. Consequently, these methods do not reveal the analytic structure of the optimal quaternion. In this work, we derive an explicit analytical characterization of all quaternions that yield zero Wahba cost for the case $\ell=2$. Our approach builds on a connection between quaternion similarity, the singular Sylvester equation $aq=qb$, and quaternion square roots established in our previous work [1]. We provide (i) necessary and sufficient conditions under which the Wahba's cost function is zero and (ii) a closed-form parameterization of all such quaternions. This eliminates the need for eigenvalue computations and enables a direct algebraic understanding of the underlying geometry of Wahba's problem.