Representing a curve contained in a surface is very important in dealing with path generation in computer numerical control (CNC) machining and the trimming issues that frequently occur in the field of CAD/CAM. This paper develops methods for tangent direction continuous (G¹) and both tangent direction and curvature continuous (G²) interpolation of a range of points on surface with specified tangent and either a curvature vector or a geodesic curvature at every point. As a special case of the interpolation, the blending problems of curves on surface are also discussed. The basic idea is as follows: with the help of the related results of differential geometry, the problem of interpolating curve on a parametric surface is converted to a similar one on its parametric plane. The methods can express the G¹ and G² interpolation curve of an arbitrary sequence of points on a parametric surface in a 2D implicit form, which transforms the geometric problem of surface intersection, usually a troublesome issue, into the algebraic problem of computing an implicit curve in displaying such an interpolation curve. Experimental results show the presented methods are feasible and applicable to CAD/CAM and Computer Graphics.