This paper presents an alternative radiosity formulation using piecewise smooth radiance functions that incorporates curved surfaces directly. Using the Galerkin integral equation technique as a mathematical foundation, surface radiance functions are approximated by polynomials. This model eliminates the need for a posteriori rendering interpolation, and allows the direct use of non-planar parametric surfaces. Convergence problems due to singularities in the radiosity kernel are analyzed and rectified, and sources of approximation error are examined. The incorporation of a shadow masking technique vastly reduces the need for meshing and associated storage space—accurate radiosity calculations can often be made with no meshing. The technique is demonstrated on traditional radiosity scenes, as well as environments with untessellated curved surfaces.