In this paper we present a nonlinear curve subdivision scheme, suitable for designing curves on surfaces. Starting with a geodesic control polygon with vertices on a surface, the scheme generates a sequence of geodesic polygons that converges to a continuous curve on the surface. In the planar case, the limit curve is a conic Bezier spline curve. Each section of the subdivision curve, corresponding to three consecutive points of the control polygon, depends on a free parameter which can be used to obtain a local control of the shape of the curve. Furthermore, it has the convex hull property. Results are extended to triangulated surfaces showing that the scheme is suitable for designing curves on these surfaces and has the convex hull property.