In this note we are concerned with the behavior of geodesies in Euclidean
n
n
-space with a smooth obstacle. Our principal result is that if the obstacle is locally analytic, that is, locally of the form
x
n
=
f
(
x
1
,
…
,
x
n
−
1
)
{x_n} = f({x_1}, \ldots ,{x_{n - 1}})
for a real analytic function
f
f
, 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
∞
{C^\infty }
boundary. Indeed, even in
E
2
{E^2}
, where our result is obvious for analytic boundaries, we can construct a
C
∞
{C^\infty }
boundary so that the closure of the set of switch points is of positive measure.