### A POV-Ray Usage Specifics

The raytracing program POV-Ray[2, 3] was used for the examples.

#### A.1 Sun Placement

The sun was set at a 3 degree horizontal and a 20 degree vertical angle from the
origin by using the commands specified in figure 8.

To calculate the unit vector 〈-0.052336,0.341551,-0.938405〉 shown in figure 8,
we let θ represent the vertical angle and we let φ represent the horizontal
angle.

From this, the following formulas represent the unit vectors used:

| (1) |

| (2) |

| (3) |

Verification that 〈x,y,z〉 is a unit vector can be accomplished with the
formula

| (4) |

Using formulas 1, 2 and 3, as well as the relationship

| (5) |

the following computations are performed:

(Note: the hard-coding of the unit vector in figure 8 must be changed in
time.)

There are extra commands available in case an area light is desired, as
documented in §A.2.

#### A.2 Area Light Computations

If soft shadows from objects distant from, say, the ground are desired, this effect is
simulated with an area light. This simulates the necessary umbra and penumbra in a
manner much like the sun.

There is a plane the area lights are attached to, so to speak. To determine which
way and which length the axis vectors u and v with a distance d are drawn are
computed with the formulas

| (7) |

and

| (8) |

The commands used to do this are shown in figure 9.

In time, more accurate vector computation routines will be determined and
used.

#### A.3 The Final Result

A picture showing a view of the “sun” approximating normal human visual angles is
shown in figure 10.

This graphic is provided to check the placement and proportions of the sun. Code
which generates this graphic is provided in figure 11.