[en] We present a new code capable of realistically modelling the radiation from ideal undulator sources, including explicit electron emittance effects. The code has been developed as part of the SHADOW X-ray optics ray tracing program, in order to predict the performances of undulator-based optical systems. The approach is based on a simplified field distribution that allows an efficient calculation of the trajectory and of the radiation field. We show explicitly that the undulator radiation pattern is shift-invariant with respect to the electron trajectory angles, so that the emittance effects can be included without recomputing the radiation distribution. Furthermore, the time consuming computation of the undulator emission patterns is decoupled from the Monte Carlo sampling of the wavefront, leading to a very fast code. This is achieved by computing the three-dimensional source probability distribution function and by using an inversion algorithm to generate a random variate with the same distribution as the source. The physical basis of the code and the algorithm used are discussed in detail and some results presented