Abstract: In this paper, we introduce a novel nonlinear curve subdivision scheme, suitable for designing curves on surfaces. The scheme is based on the concept of geodesic conic Bézier curves, which represents a natural extension of geodesic Bézier curves for the rational quadratic case. Given a set of points on a surface S, the scheme generates a sequence of geodesic polygons that converges to a continuous curve on S. If the surface S is C$^2$-continuous, then the subdivision curve is C$^1$-continuous and if S is a plane, then the limit curve is a conic Bézier spline curve. Each section of the subdivision curve depends on a free parameter that may be used to obtain a local control of the shape of the subdivision curve. Extending these results to triangulated surfaces, it is shown that the scheme has the convex hull property and that it is suitable for free-form design on triangulations.