A novel technique that uses the Moving Least Squares (MLS) method to interpolate sparse constraints over mesh surfaces is introduced in this paper. Given a set of constraints, the proposed technique constructs, directly on the surface, a smooth scalar field that interpolates or approximates the constraints. Three types of constraints: point-value, point-gradient and iso-contour, are introduced to provide flexible control of the scalar field design. Furthermore, the framework also provides the user control over the region of influence and rate of influence decrease of the constraints, through parameter adjustment. We demonstrate that the scalar fields resulted from this framework can be used in several mesh applications such as skin deformation, region selection and curve drawing.