New Mathematical Method for Computer Graphics

Rendering two-dimensional data in the case of rough, complex surfaces is a challenge in computer graphics. Typically, splines with knots and control points are used, and while they yield useful surfaces they can be of poor quality, or can be difficult to apply. Fundamentally splines are local. An alternative mathematical method is to construct global methods which can be tuned to have polynomial behavior, or behave in ways that are not as restrictive, and which can be local, or not, depending on user input. This study examines a hybrid method of stochastic interpolation built around Bernstein functions. This approach is non-polynomial and global, but readily computable and can successfully fit complex two-dimensional surface data to obtain high quality at low computational cost. The representation of parametric surfaces in 3 dimensions can be achieved using approximation, or interpolation using this method. The generation of computational surfaces rendered using OpenGL, shows that this hybrid method of Bernstein function interpolation is a sound approach to surface rendering, and computational issues in achieving speed with accuracy are discussed. The hybrid method is shown to be robust, and can be selectively adjusted to yield controlled smoothing of the surface data. The method enables use of computational stencils of arbitrary size, and permits the construction of infinitely differentiable surfaces if needed.