  This paper is concerned with rational curves on real classical groups. Our
contributions are three-fold: (i) We determine the structure of quadratic
rational curves on real classical groups. As a consequence, we completely
classify quadratic rational curves on $\mathrm{U}_n$,
$\mathrm{O}_n(\mathbb{R})$, $\mathrm{O}_{n-1,1}(\mathbb{R})$ and
$\mathrm{O}_{n-2,2}(\mathbb{R})$. (ii) We prove a decomposition theorem for
rational curves on real classical groups, which can be regarded as a
non-commutative generalization of the fundamental theorem of algebra and
partial fraction decomposition. (iii) As an application of (i) and (ii), we
generalize Kempe's Universality Theorem to rational curves on homogeneous
spaces.
