In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique Thompson's metric geodesic connecting $$x$$ and $$y$$ in the cone of positive self-adjoint elements in a unital $$C^*$$-algebra if, and only if, the spectrum of $$x^{-1/2}yx^{-1/2}$$ is contained in $$\{1/\beta,\beta\}$$ for some $$\beta ≥ 1$$. A similar result will be established for symmetric cones. Secondly, it will be shown that if $$C^°$$ is the interior of a finite-dimensional closed cone $$C$$, then the Thompson's metric space $$(C^°,d_C)$$ can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, $$C$$ is a polyhedral cone. Moreover, $$(C^°,d_C)$$ is isometric to a finite-dimensional normed space if, and only if, $$C$$ is a simplicial cone. It will also be shown that if $$C^°$$ is the interior of a strictly convex cone $$C$$ with $$3 ≤ dim \space C ≤ \infty$$, then every Thompson's metric isometry is projectively linear.