The previous chapters explored two approaches to the task of rendering four-dimensional images: wireframe display and raytracing. Both techniques have advantages and disadvantages over the other; e.g. wireframe display is the only real solution to rendering four-space curves. It also allows for rapid display of a four-dimensional structure.
Raytracing, on the other hand, allows the user to view surfaces and solids in and of four dimensions. It also provides other important visual cues, such as shadows, highlights, and reflections. In addition, the output images make it clear which parts are solids projected from four-space; the wireframe approach is subject to ambiguity in the projected image.
This research began with the goal of visualizing four-dimensional structures in four dimensions. While several techniques exist (and many more are currently being developed) to visualize four dimensional data as 3D scalar fields, there are few techniques that exist to visualize four-space geometry.
There are, in fact, several 4D wireframe display programs; the earliest documented was written around 1967. The wireframe display program presented in this paper combines the wireframe display with the viewing model presented in [Foley 87], which is a simple and efficient method of projection. In addition, the program written for this research allows for the 4D depth-cueing of the display data, the interactive manipulation of the 4D object, and the interactive selection of the projection modes.
The most promising field of application for this research is the field of Computer-Aided Geometric Design, for the use of displaying curves and surfaces in four dimensions. The wireframe viewer has been used to view 4D spline curves and has displayed artifacts that were not obvious with other methods (see figure 4.15).
The raytracer written for this research implements the four-sphere, the four-tetrahedron, and the four-parallelepiped. It handles point & directional lighting, reflection, refraction, plus ambient, diffuse and specular lighting.
The primary catch with four-dimensional raytracing is the fact that the resulting image is a three-dimensional voxel field, which (for ``interesting'' images) will have a complex internal structure that is difficult to visualize with current techniques.
There's a lot of room for expansion of the 4D raytracer. One obvious area is the inclusion of additional fundamental objects for the raytracer. As mentioned earlier, all 4D objects can be represented with a mesh of tessellating tetrahedra, but this is quite expensive in terms of both storage and time. All that is really needed for a new four-dimensional object is an implicit equation of its hypersurface. The four-dimensional ray equation can be plugged into the implicit object equation to yield the equation for the intersection points. In the case of multiple intersections, the closest intersection point is selected.
In addition, the display of the resulting voxel field could well bear some research. Most visualization techniques work on a 3D space of scalar data; it would be useful if some techniques existed to display a 3D field of RGB data.
The voxel field generated by the raytracer is somewhat different from other fields more often associated with four-dimensional visualization, which are often amorphous fields of scalar values. The output voxel fields of the raytracer are characterized by the following properties:
In order to further understand the 4D images, stereo display techniques for both the wireframe display and the raytrace output may prove useful. There are problems with stereo displays of higher dimensions, primarily the extra degree of parallax, but there may be ways to solve these. See [Brisson 78] for an example of 4D stereograms.
| Previous Chapter | References | Appendix A | Back To Table of Contents |