Graphics Reference
In-Depth Information
Figure 13.2. The region of space that would be visible through a reflective paraboloid.
this is similar to viewing your surroundings through a Christmas tree bulb—you will see a
distorted view of the objects that are in the room with you. However, a paraboloid is shaped
differently than a spherical bulb, and hence will produce a different view of the objects
around it. To better characterize what would be visible in this surface, we will examine
some of the basic properties of the paraboloid surface. We begin with the mathematical
definition of the paraboloid surface itself, which is presented in Equation (13.1):
(13.1)
The surface that results from evaluating this equation over an x- and y-domain of
[-1,1] is depicted in Figure 13.1. In this figure, we see that the surface has a maximum
when x and y are zero, and the value falls off as x and y move away from zero. We will be
interested in the region of this surface where the result of this equation is positive, meaning
that the sum of the square of x and y must be less than or equal to 1.
Returning to our initial consideration of the reflective surface, we can see that if we
view the paraboloid from above over the domain [-1,1], the reflections will vary over the
hemisphere above its zero plane. This region is depicted in Figure 13.2 by drawing arrows
from the viewing direction and seeing the direction that they would be reflected toward.
This ability to "see" an entire hemisphere is a very important property, since it means
that we can use a single paraboloid projected onto a plane to capture precisely one half
of the surroundings of a particular point in space. With our viewing direction from above
the paraboloid, we can visualize this projection of the paraboloid onto a viewing plane
between us and the paraboloid. This is roughly depicted in Figure 13.3.
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