  Many applications of geometry modeling and computer graphics necessite
accurate curvature estimations of curves on the plane or on manifolds. In this
paper, we define the notion of the discrete geodesic curvature of a geodesic
polygon on a smooth surface. We show that, when a geodesic polygon P is closely
inscribed on a $C^2$-regular curve, the discrete geodesic curvature of P
estimates the geodesic curvature of C. This result allows us to evaluate the
geodesic curvature of discrete curves on surfaces. In particular, we apply such
result to planar and spherical 4-point angle-based subdivision schemes. We show
that such schemes cannot generate in general $G^2$-continuous curves. We also
give a novel example of $G^2$-continuous subdivision scheme on the unit sphere
using only points and discrete geodesic curvature called curvature-based
6-point spherical scheme.
