.CH "Conclusion" 6
.PP
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; \fIe.g.\fR wireframe display is the
only real solution to rendering four-space curves.  It also
allows for rapid display of a four-dimensional structure.
.PP
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.
.SE "Research Conclusions" 1
.PP
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.
.PP
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.
.PP
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).
.PP
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.
.PP
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.
.SE "Future Research Areas" 2
.PP
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.
.PP
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.
.PP
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 1c
.LP 1)
Internal boundaries are well-defined, corresponding to projected objects.
.LP 2)
There can be quite a lot of different internal solids, often intersecting.
.LP 3)
Each voxel is assigned an RGB triple.
.in 0
.sp 1.5v
.PP
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.
