  Given any arbitrary semi-algebraic set $X$, any two points in $X$ may be
joined by a piecewise $C^2$ path $\gamma$ of shortest length. Suppose
$\mathcal{A}$ is a semi-algebraic stratification of $X$ such that each
component of $\gamma \cap \mathcal{A}$ is either a singleton or a real analytic
geodesic segment in $\mathcal{A}$, the question is whether $\gamma \cap
\mathcal{A}$ has at most finitely many such components. This paper gives a
semi-algebraic stratification, in particular a cell decomposition, of a real
semi-algebraic set in the plane whose open cells have this finiteness property.
This provides insights for high dimensional stratifications of semi-algebraic
sets in connection with geodesics.

Algebraic Geometry;Geometric Topology