Geodesic through/between singularities

Question

Let $X\subseteq\mathbb R^q$ be a (singular) real algebraic set, and let $$ g\colon[0,T]\to X $$ be a geodesic (that is, a shortest path between its ends). Is it true that the image $g([0,T])$ is contained in some (possibly singular) analytic curve? By an analytic curve I mean a 1-dimensional analytic subset of an open set in $X$.