This paper builds two detailed examples of generalized normal in
non-Euclidean spaces, i.e. the hyperbolic and elliptic geometries. In the
hyperbolic plane we define a n-sided hyperbolic polygon P, which is the
Euclidean closure of the hyperbolic plane H, bounded by n hyperbolic geodesic
segments. The polygon P is built by considering the unique geodesic that
connects the n+2 vertices (tilde z),z0,z1,…,z(n-1),z(n). The geodesics that
link the vertices are Euclidean semicircles centred on the real axis. The
vector normal to the geodesic linking two consecutive vertices is evaluated and
turns out to be discontinuous. Within the framework of elliptic geometry, we
solve the geodesic equation and construct a geodesic triangle. Also in this
case, we obtain a discontinuous normal vector field. Last, the possible
application to two-dimensional Euclidean quantum gravity is outlined.

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