Let's have a brief discussion on geometry and trigonometry.  I know
you are _probably_ thinking, "Ick."  But this is necessary if you are
to understand just how traveling through space is going to work
while navigating a space ship.

First of all, some terminology :             Z
                                             | _  90
   X : The X coordinate plane/axis           |   \
   Y : The Y coordinate plane/axis           |____|___ X
   Z : The Z coordinate plane/axis          /
                                         Y/

At any time you might imagine your ship at the origin, or center, of
these three axes.  Heading, or in which direction you would like
to travel, is specified in two parts:  head <XY> mark <Z>

The first part is the XY angle, or the two dimensional angle within the
XY plane where you would like to head.  This can be pictured as
a traditional flat map, such as a map of the U.S.

When thinking of a 2D map, a small conflict sometimes arises.
Trigonometry courses teach that 0 and 180 degrees are on the X axis,
while 90 and 270 are on the Y.  However, Earth directions are
specified in bearings.  More currently, they are specified in Azimuths.
When using Azimuth, which the space navigation system does, the angles
get changed slightly as follows:
                                        North
      090                                000
       |                                  |
180 ---|--- 000               West 270 ---|--- 090 East
       |                                  |
      270                                180
                                        South

 (Standard Trigonometric Axes)     (The Azimuth System)

The second part of the navigational heading command is the Z heading,
or Z vector specification.  This is a number between -90 and 90
degrees with -90 pointing directly "down" and 90 pointing directly
"up."  The command "head 180 mark 90" would turn the ship to the
south, but pointing directly up the Z axis.  Once thrust is applied,
only the Z coordinate of the ship would change.

Now for an example of travel.  Say, for instance, my ship is located
at 0, 0, 0.  That is 0 X, 0 Y, and 0 Z - I am at the origin of all
axes.  I am aware of another ship that is located very nearby at
1, 1, 1.  If I am to intercept that ship, where do I head?  Rather
than telling the computer that I want to end up at 1, 1, 1; let's
imagine that I have a pilot's flight stick in front of me.  This will
maneuver my ship to its destination.  Using my calculator (not really),
I see that, on the XY plane, the ship is at a bearing of 45 degrees
from me:  atan(1/1) = 45.
                           |   . <- enemy
                           |  / <---------- 45 Degrees (XY plane)
                    X ---- Me ---
                           |
                           |
                           Y

That fills the first part of the two-part information I need to set
my heading.  Next, I need to know the Z angle to specify.  This is,
in fact, simple.  It is also 45 degrees.  Why?  I want to travel
equal amounts on each axis.  On the XY plane, 45 degrees is halfway
between the Y axis and X axis.  Likewise, 45 degrees on the Z plane
is halfway between up and level.

                 Z (90 degrees)
                 |   . <- Enemy
                 |
                Me ------- XY plane (Z angle of 0)

There, we have our heading: head 45 mark 45

We can see that if our destination were 0, 0, 1; we would need to
ONLY travel on the Z axis.  Therefore, an XY heading is irrelevant,
and the Z angle is the only importance.  To travel straight "up," we
would need a Z angle of 90 degrees: head <anything> mark 90.

Movement on all three axes is calculated by simply using the sine
and cosine functions as follows:

       dz = sine(z heading)
       dx = sine(xy heading) * cosine(z heading)
       dy = cosine(xy heading) * cosine(zheading)

We see that, as the z heading increases, the cosine of the z heading
decreases.  Thus, this results in decreased travel on the XY axes.

Calculating a heading from one location to another is rather simple
and involves the use of two provided functions: xyang() and zang().
As you might guess, xyang() provides the XY part of the heading; and
zang() provides the Z part of the heading.  They can be used as
follows:

  XY = xyang(current x, current y, desired x, desired y)
  Z  = zang(current x, current y, current z, desired x, desired y,
            desired z);

head XY mark Z

For some, the navigational system seems overwhelming at first.  For
others, it comes very quickly.  It all depends on how the individual
pilot visualizes his/her surrounding space.  Often, it helps to pull
out a few pieces of graph paper and draw a picture to determine where
it is that you are and where it is that you want to go.

In time, though, navigation becomes a simple fact-of-the-matter.
Headings are quickly plotted and missiles outmaneuvered.  In time,
you too can become an ace pilot.