The special Euclidean group \(\textrm{SE}_3\) and the closely related group \(\textrm{SO}_3\times \mathbb {R}^3\) are important spaces in many fields of application. Explicit embeddings into matrix subgroups are being studied and it is shown that these spaces are extrinsic symmetric with respect to their embeddings. Furthermore, endpoint geodesic formulas are derived explicitly in both cases, as well as for certain subspaces of \(\textrm{SE}_3\) . Ultimately, the formulas could be used to solve for geodesics that connect two given points on these spaces.
