# Mathematical Combat Distance

**1/27/07**

Intuitively, one understands that danger is inversely proportional to the distance from a threat. In the martial arts, this concept of "combative distance" is crucial.

This informal article will outline one way of looking at combat distance. I've personally found it useful, and hope others can.

Let's represent each Person_{i} by a circle of radius Leg_{i} and concentric circle of radius Arm_{i}. The center of the circle is the center of a person; their center of gravity, core, tanden, tantien, etc.

Each radius is the reach of the arm and leg respectively, and the extent of each reach is marked by the edge of the circle. Note that these distances can be measured for any person. Simply stand stationary, extend a limb, and have someone measure the distance the limb extends, using a measuring tape.

What happens when two people, who are a distance, d, apart, approach each other? The outcome (keep in mind our assumptions, listed at the end of the article), hinges on two factors

- How does Leg
_{1}compare to Leg_{2}? - How does Arm
_{1}compare to Arm_{2}?

If these people approach (cross your eyes to make them approach), Person_{1} is able to touch Person_{2}'s center first or able to "cover" a larger fraction of Person_{2}. How can we mathematically express this?

Note that when Person_{1} is able to do this, more of Person_{2} is overlapped by Person_{1} than vice versa. This overlap, expressed mathematically as a measure of area, is

_{1}+Leg

_{2})(d+Leg

_{1}-Leg

_{2})(d-Leg

_{1}+Leg

_{2})(d+Leg

_{1}+Leg

_{2})]

^{1/2}

Therefore, the percentage overlap of Person_{2} is the overlap divided by Person_{2}'s area, or

_{1}+Leg

_{2})(d+Leg

_{1}-Leg

_{2})(d-Leg

_{1}+Leg

_{2})(d+Leg

_{1}+Leg

_{2})]

^{1/2})) / (pi*Leg

_{2}

^{2})

Let's label this number by LegDanger_{2}, and note that if LegDanger_{i} > LegDanger_{j}, then Person_{i} is in more trouble than Person_{j} from leg attacks.

Note, however, that if Person_{1}'s leg attack fails, then Person_{2}'s arm attack is able to get to Person_{1}'s center first, barely. This is why one must take into account both leg and arm attacks. The above formula for A is calculated using the Arm radii, and we get ArmDanger_{1} and ArmDanger_{2} respectively.

Therefore, Person_{1}'s total danger is

_{1}= LegDanger

_{1}+ ArmDanger

_{1}

Written out in full, this is

_{1}= [(((1/2) * [(-d+Leg

_{1}+Leg

_{2})(d+Leg

_{1}-Leg

_{2})(d-Leg

_{1}+Leg

_{2})(d+Leg

_{1}+Leg

_{2})]

^{1/2})) / (pi*Leg

_{1}

^{2})] + [(((1/2) * [(-d+Arm

_{1}+Arm

_{2})(d+Arm

_{1}-Arm

_{2})(d-Arm

_{1}+Arm

_{2})(d+Arm

_{1}+Arm

_{2})]

^{1/2})) / (pi*Arm

_{1}

^{2})]

For a given time in an encounter, one can compare Danger_{1} and Danger_{2}. If Danger_{1} > Danger_{2}, then Person_{1} is, theoretically, in trouble.

The discussion has been held to open-hand combat. Most weapons could be included in the Arm radius. However, a weapon like a gun would have a huge radius, hundreds of feet, and its circle would easily overlap any non-weapon carrying person.

What happens with multiple opponents? Say Person_{1} is being attacked by Person_{2} and Person_{3}

To assess danger, we'd compare Danger_{1} to Danger_{2} + Danger_{3}.

In this article, there are several assumptions that are needed so we can talk about such a simple model

- model is 2-dimensional. One could consider doing the same analysis using spheres and volumes
- each person can attack and defend 360 degrees
- beside arm and leg reach, each person is equal in all other aspects
- attacking or controlling the opponent's center is the goal, and is most devastating

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