In this paper, we introduce a general-purpose method for computing radiosity on scenes made of parametric surfaces with arbitrary trimming curves. By contrast with past approaches that require a tessellation of the input surfaces (be it made up of triangles or patches with simple trimming curves) or some form of geometric approximation, our method takes fully advantage of the rich and compact mathematical representation of objects. At its core lies the \emph{virtual mesh}, an abstraction of the input geometry that allows complex shapes to be illuminated as if they were simple primitives. The virtual mesh is a collection of normalized square domains to which the input surfaces are mapped while preserving their energy properties. Radiosity values are then computed on these supports before being lifted back to the original surfaces. To demonstrate the power of our method, we describe a high-order wavelet radiosity implementation that uses the virtual mesh. Examples of objects and environments, designed for interactive applications or virtual reality, are presented. They prove that, by exactly integrating curved surfaces in the resolution process, the virtual mesh allows complex scenes to be rendered more quickly, more accurately and much more naturally than with previously known methods.