The absolute orientation problem arises often in vision and robotics. Despite that robust algorithmic solutions exist for quite some time, they all rely on matrix factorizations such as eigen or singular value decomposition. These factorizations are relatively expensive to compute, therefore might become a performance bottleneck when absolute orientation needs to be repeatedly computed on low-end hardware. The issue is exacerbated by implementations relying on standard numerical software libraries like LAPACK, since the linear algebra factorization routines they include are optimized for large matrices and thus are not the most efficient choice for small ones. Based on an attitude estimation algorithm originating from astronautics, this paper proposes a direct, factorization-free solution to the absolute orientation problem that is both computationally efficient and numerically accurate. Results from an experimental comparison with established methods demonstrate its superior performance.