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This paper concerns an analytical stratification question of real algebraicand semi-algebraic sets. For Whitney's stratification in 1957, it partitions areal algebraic set into partial algebraic manifolds\cite{W}. In 1975 Hironakareproved that a real algebraic set is triangulable and also generalized it tosemi-algebraic sets, following the idea of Lojasiewicz's triangulation ofsemi-analytic sets in 1964. Following their examples and wondering how geometrylooks like locally. this paper tries to come up with a stratification, inparticular a cell decomposition, such that it satisfies the followinganalytical property. Given any shortest curve between two points in a realalgebraic or semi-algebraic set, it interacts each cell (or simplex) at mostfinitely many times.

A triangulation of semi-algebraic sets concerning an analytical condition for shortest-length curves