Abstract: The theory of the multivariate polyhedral splines is applied to analytic antialiasing: a triangular simplex spline is used to represent surface intensity, while a box spline is used as a filter. Their continuous convolution is a prism spline that can be evaluated exactly via recurrence. Evaluation performance can be maximized by exploiting the properties of the prism spline and its relationship to the sampling grid. After sampling, digital signal processing can be used to evaluate exactly and efficiently the sampled result of any analytic spline filter in the span of the box spline basis used as the original analytic filter. Our result advances the state of the art by providing not only high quality analytical antialiasing but also a connection to a variety of useful spline techniques. Polyhedral spline techniques can also potentially be extended to other rendering tasks that require multivariate integration.