Introduction

High level mathematical functions such as the Bessel functions, the gamma and beta functions, hypergeometric functions and others are important for solving many problems in the mathematical and physical sciences. For example, the Airy functions Ai and Bi, which are Bessel functions of fractional order, provide closed form solutions to field equations that arise in quantum mechanics, optics and electromagnetism. The gamma and beta functions provide the starting point for the computation of more complex functions such as the Riemann zeta function and others that occur in number theory and mathematical physics. Because of their importance, references which discuss the definition and computations of these ``special functions'' continue to be widely used. One such reference is the National Bureau of Standards (NBS) Handbook of Mathematical Functions [1]. Its popularity has led the National Institute of Standards and Technology (NIST), the successor organization to NBS, to begin a large scale project to update and expand the handbook and disseminate it on the World Wide Web as the NIST Digital Library of Mathematical Functions (DLMF). A key feature of the DLMF will be 3D graphics and visualization capabilities that allow a user to interactively examine the unique features of complicated mathematical functions.

The complex nature of these functions makes the visualization task quite difficult. The presence of singularities and poles usually means that the computational domain will be irregular, discontinuous, or multi-connected. This paper discusses the use of grid generation techniques to facilitate the plotting of surfaces that are the graphs of functions, that is, surfaces that can be described by equations of the form $z=f(x,y)$. Some commercial packages have a few of the functions built in and may allow the user to produce a plot, usually over a rectangular cartesian mesh. However, this often produces a very poor and in many cases misleading graph of the function. In addition, the packages often have problems clipping the surface properly when values fall outside the range of interest specified by the user. Furthermore, we have often found that what looks satisfactory inside the package, may not when we transform the data to a format more suitable for interactive graphics on the Web.

We are looking at various techniques for solving the problems we have encountered using commercial packages. These include using a computational grid whose boundary coincides with the contours of the surface, adapting the grid lines to obtain more concentration in areas of large curvature, or designing the entire coordinate system so that the grid lines correspond to contours of the surface and the curves orthogonal to the contours. This paper discusses the grid generation techniques that have been tried to date and in particular looks at the effectiveness of an updated version of a tensor product B-spline grid generation algorithm designed by one of the authors [2]. The feasibility of using unstructured techniques and the affect of their use on the translation of the data to other formats are also discussed.