For most compute environments, adaptive forward differencing is much more efficient when performed using integer arithmetic than when using floating point. Previously low precision integer methods suffered from serious precision problems due to the error accumulation inherent to forward differencing techniques. This paper proposes several different techniques for implementing adaptive forward differencing using integer arithmetic, and provides an error analysis of forward differencing which is useful as a guide for integer AFD implementation. The proposed technique using 32 bit integer values is capable of rendering curves having more than 4K forward steps with an accumulated error of less than one pixel and no overflow problems. A hybrid algorithm employing integer AFD is proposed for rendering antialiased, texture-mapped bicubic surfaces.

Trimmed non-uniform rational B-splines have become a very useful surface representation form in the mechanical CAD industry. Previous rendering methods use the de Boor algorithm to evaluate the surface at equal increments in parameter space. This yields polygons which are then rendered. Alternatively the Oslo algorithm and Boehm's knot insertion algorithms are used in a subdivision approach. In this paper a new method is presented for rendering trimmed NURB surfaces of arbitrary order using the adaptive forward differencing (AFD) technique. This method extends the AFD technique to higher order, efficiently computes the basis matrix for each span, calculates the shading approximation functions for rational surfaces, and trims and image maps NURB surfaces. Trimming is accomplished by using AFD to scan convert the trimming curves in parameter space, thus producing the intersection points between the trim curves and an isoparametric curve across the surface. A winding rule is used to determine the regions bounded by the curve which are then rendered with AFD. The method is suitable for both hardware and software implementations, however, higher order surfaces require very high precision due to the forward difference nature of the algorithm.