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In this note we are concerned with the behavior of geodesies in Euclidean «-space with a smooth obstacle. Our principal result is that if the obstacle is locally analytic, that is, locally of the form xn = f(xx, ... , xn_x) for a real analytic function /, then a geodesic can have, in any segment of finite arc length, only a finite number of distinct switch points, points on the boundary that bound a segment not touching the boundary. This result is certainly false that for a C°° boundary.doi:10.2307/2048459 fatcat:oipxptvf4ngy7l55jbuf4q5ssq