# Thanks, Pythagoras:

## How To Visualize Three Dimensions On Your Two-Dimensional
Map

**by Captain Mark R. Smith**
After concluding my enemy situation brief, my
battalion commander asked me about the dimensions of the engagement
area on the ground of a specific antiaircraft gun system if the gun
was fully depressed below the horizon. After I recovered from the
"deer in the headlights stare," I told him frankly that I did not
know, but that I would figure it out, wistfully hoping he would not
take me up on it. As expected, he told me that was a fine idea, "Do
so."
Since I did not have any of the fancy software-based terrain
analysis tools, I was forced to use the stubby-pencil method. I
remembered only too well that math, let alone geometry, was not my
strong suit in high school. Nevertheless, I rolled up my sleeves
and began going over the principles I had learned in high school
geometry to turn the two-dimensional map in front of me into a
three-dimensional tool.
## Pythagorean Theorem

What came to mind was a theorem the Greek philosopher and
mathematician Pythagoras (or one of his students) had proven in the
6th century B.C.E. The theorem states that-

**AA + BB = CC**

In other words, the square of the hypotenuse of a right triangle is
equal to the sum of the squares of its other two sides.1 Knowing
this, I could use the maximum effective range of the weapon system,
and the system's altitude (assuming the weapon is set up
perpendicular to the ground) to figure out the base of this right
triangle and determine the maximum effective range on the ground.
I manipulated the theorem to:

AA (altitude/height) + BB (base/length) = CC (hypotenuse/maximum
effective range) to **CC - AA = BB.**
## Minimum Effective Range

Once I had determined the weapon's maximum effective range, I could
figure out its minimum effective range. For this task, I needed a
protractor and to know the weapon system's maximum depression below
the horizon.
I measured the angle formed by the height (altitude) and the
hypotenuse (maximum effective range). Then I compared the angle to
the maximum depression below the horizon. If the angle was not at
its maximum depression below the horizon, I redrew the angle and
the hypotenuse to the maximum depression below the horizon. I thus
established the minimum effective range of the weapon system and
figured out the engagement area of the weapon system on the ground.
If the angle was already at its maximum depression below the
horizon, then I knew that the maximum and minimum effective ranges
were the same. If the angle was below the weapon system's maximum
depression below the horizon, then I knew that the weapon system's
maximum and minimum effective range would not be on the ground.
Once I figured out the weapon system's engagement area, I then
determined the key to defeating or avoiding the weapon system its
dead space.
## Effects of Altitude

During my calculations I realized that the altitude of the weapon
system affected the size of the weapon system's engagement area
(see Figure 1). I noted that the higher the weapon system's
altitude, the smaller the engagement area, and the larger the dead
space. However, the higher the altitude of the weapon system, the
lower the probability of plunging fire being degraded by obstacles
between the weapon system and the target.
## Conclusion

Knowing the enemy engagement area of any direct fire system allows
me to brief the commander on how to optimize his use of the
terrain. Knowing the enemy engagement area also aids the warfighter
in defeating the weapon system by capitalizing on the weakness of
the weapon system, its dead space, and turning the weakness into a
vulnerability.

Thanks, Pythagoras.
##### Endnote:

1. Bruce E. Meserve, Grolier Electronic
Publishing, Inc., 1992.
*Captain Smith is the S2, 1st Battalion, 503d Infantry
Regiment (Air Assault), 2d Infantry Division, at Camp Casey,
Republic of Korea. He will complete his present tour in February
1996 with a follow-on assignment to the XVIII Airborne Corps, Fort
Bragg, North Carolina.*