3D Visualization in a Web-Based Digital Library

For the interactive visualizations in the DLMF we begin with a preprocessing stage, using available packages such as MATLAB, MAPLE and MATHEMATICA to plot the data so that we can examine the graphical representation and adjust the scaling to bring out interesting features. The data is then converted to VRML (Virtual Reality Modeling Language) format. VRML [3] is a standard 3D file format for describing the behavior and geometry of a 3D virtual world, or scene. Its accessibility on the Internet and interactive capabilities make it an ideal candidate for this development work. It is not a foregone conclusion that the final version of the DLMF will use VRML. This may depend on whether VRML browsers continue to be readily available. We are looking at alternatives to VRML such as JAVA 3D which would not require the download of a browser, but still would require the user to obtain the graphics package. Another choice may be X3D, an emerging next-generation technology which will extend the capabilities of current VRML. In the mockup DLMF already developed and located at http://dlmf.nist.gov/, the user has the option of viewing a still 3D image if a VRML browser is not available. Figure 1 shows a VRML display from the prototype chapter on Airy functions in the mockup Web site. The display shows $\vert\mbox{Ai}(z)\vert$ in a CosmoPlayer browser. VRML browser controls allow the user to rotate the figure, zoom in and out, and move the figure in an arbitrary direction. We have added custom vcr type controls that let the user move a cutting plane through the surface and observe the motion of the intersection curve.

Figure 1: VRML display on CosmoPlayer.

When investigating commercial packages we were surprised to discover that many do not perform 3D clipping properly when points fall outside the plotting range. In some cases the default method of clipping is to reset values outside the plotting range to the same constant. This produces the misleading shelf effect seen in the Mathematica plot of $\vert\mbox{Bi}(z)\vert$ over an equally spaced rectangular domain in Figure 2. This technique is extensively used by William J. Thompson in Atlas for Computing Mathematical Functions [4].

Figure: Clipped version of $\vert\mbox{Bi}(z)\vert$ using Mathematica.

By computing the function over a grid whose boundary matches a contour of the function, this problem can be eliminated. Also, the contour fitted grid tends to produce a smoother shading when the data is translated to VRML format.