We present an approach to surface smoothing based on linear diffusion. In contrast to mesh-based approaches we represent surfaces as scalar functions defined on the sphere which restricts the analysis to star-shaped objects. We derive the Green?s function of the linear diffusion process on the sphere and identify it with the spherical Gauss function. This allows us to introduce a linear scale-space or data on the sphere. The approach is computationally efficient since does not require iterative smoothing. Several examples are presented demonstrating the evolution of surfaces and their parabolic lines under the spherical diffusion process.