FDGE: A forward-difference grid engine and its application in rendering algebraic surfaces in parallel

The display of an implicitly defined surface is obtained by its projection onto the viewing plane. Ray casting is a technique that accomplishes this projection byfiring a mapping ray through each pixel of the screen into the world space. The intersection points of this ray with the surface are found; these points are further tested to determine which one is visible and is within the viewing volume. Finally, if a point that satisfies the above conditions is found, then the point is further processed for the determination of its shading value.Much progress in rendering algebraic surfaces has been made recently. However, most of the proposed solutions are based on subdivision methods. This paper focuses on a method that performsdirect rendering of such a surface that will minimize the number of operations; this method will also replace almost all required multiplications with additions. Furthermore, the VLSI-oriented algorithm lends itself to parallelism. In addition, how this method can be used in the calculation of the shading values is described and a VLSI architecture is briefly discussed. Finally, a by-product of this method is that it can be used to efficiently calculate the values of a bivariate polynomial in a rectangular grid and in parallel.