The problem of constructing curve on parametric surface (or surface that can be parameterized) such that it interpolates a sequence of points with prescribed tangent direction and curvature vector (or geodesic curvature) at every point and the issue of curve blending on this kind of surface are researched. The mapping and tangent mapping from the surface to its parametric plane are introduced and thus several conclusions with differential geometry are deduced. Based on those conclusions, the problem of interpolating (or blending) curve on a parametric surface is converted to a similar one on its parametric plane. The final solution curve of either interpolation or blending issue is explicit and can still be expressed by parametric form. And so, unlike existing methods, the presented method needs not to use any surface/surface intersection algorithms, usually a troublesome process, for displaying such interpolation curve. Experiment results show the presented methods are feasible and applicable to CAD/CAM and computer graphics.