  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
$\beta\geq 1$. A similar result will be established for symmetric cones.
Secondly, it will be shown that if $C^\circ$ is the interior of a
finite-dimensional closed cone $C$, then the Thompson's metric space
$(C^\circ,d_C)$ can be quasi-isometrically embedded into a finite-dimensional
normed space if, and only if, $C$ is a polyhedral cone. Moreover,
$(C^\circ,d_C)$ is isometric to a finite-dimensional normed space if, and only
if, $C$ is a simplicial cone. It will also be shown that if $C^\circ$ is the
interior of a strictly convex cone $C$ with $3\leq \dim C<\infty$, then every
Thompson's metric isometry is projectively linear.
