# 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.