Exact evaluation of limits and tangents for non-polynomial subdivision schemes

In this paper, we describe a method for exact evaluation of a limit mesh defined via subdivision and its associated tangent vectors on a uniform grid of any size. Other exact evaluation technique either restrict the grids to have subdivision sampling and are, hence, exponentially increasing in size or make assumptions about the underlying surface being piecewise polynomial (Stam's method is a widely used technique that makes this assumption). As opposed to Stam's technique, our method works for both polynomial and non-polynomial schemes. The values for this exact evaluation scheme can be computed via a simple system of linear equation derived from the scaling relations associated with the scheme or, equivalently, as the dominant left eigenvector of an upsampled subdivision matrix associated with the scheme. To illustrate one possible application of this method, we demonstrate how to generate adaptive polygonalizations of a non-polynomial quad-based subdivision surfaces using our exact evaluation method. Our tessellation method guarantees a water-tight tessellation no matter how the surface is sampled and is quite fast. We achieve tessellation rates of over 33.5 million triangles/second using a CPU implementation.