Many of my students use to shudder when I explained that trigonometry is used in Kempo. “How can that be?” they would protest, “This is karate, not math.” To me, martial arts are applied mathematics. Just as math has “laws” and “theorems”, so to does Kempo. In fact (according to several masters), the translation of Kempo is “law of the fist”. Not the judicial law, rather it’s the biomechanical law, the laws of applied mathematics.

This article is not a dissertation on the subject; it is a quick explanation of my understanding of this concept. Joint locks require both correct (right) angles and 90° (right) angles for maximum effect. Takedowns and other body manipulation should make use of the X, Y, and Z-axis.

What I have noticed in several techniques is the right angle in controlling locks. Be they arm locks, joint freezing or body checks, the right angle hides in the technique. For the sake of brevity, we’ll use one example.

**An Example**

In overhead club defense #5, there is a T-bar lock on your opponent’s arm that requires three right angles. The first is shoulder. The second is the elbow and the final angle is the wrist. Once you lock in these angles, you can leverage the arm over easily.

In this example, if you allow one of these angles to become greater or less than 90°, it allows the opponent to slip out. The elbow is especially important. If the elbow lock is less than 90°, then the take down maneuver is very difficult to perform.

**The X, the Y and the Z**

Two-dimensional space can be organized using two axis; the X and the Y. if you’re unfamiliar with this concept, don’t worry. On a piece of graph paper, draw a vertical line and a horizontal line. They should share one point, which makes them look like an oversized “L”. The vertical line is named Y, and the horizontal line is named X.

Where the two lines meet is called the origin. Each square you move away from the origin is one step away. The Y and the X record this information for you. If you move two squares up the vertical line, you move to the two spot of Y. If you move three squares across the horizontal line, you move across to the three spot of X. To note this information, we use the structure (X, Y). This makes our example equal to (3,2).

Now it gets complicated. There is a third axis that runs into the origin point. We call this axis Z. however; we can’t draw the Z-axis on the graph paper because it sticks out of the paper. Take you pencil and place the point at the origin point. Make sure the pencil is straight up. This is the Z-axis.

To note things on this axis, we add another number to our list. The first square on the Z-axis is 1. This makes our continuing example (3,2,1).

Unfortunately, you’ll need a good trigonometry book to teach you more. I highly recommend browsing through old math books to keep you skills sharp. You never know what you’ll remember. Hopefully, this is enough to understand what I’m saying about the three axis.

**“Zero”-ing Out the Axis**

To “zero” out an axis, you eliminate diagonal pulls. This brings the coordinates of an axis to 0. If you imagine your opponent having the X, Y, and Z lines sticking out of their body, you can understand this concept. Where the lines meet, in the middle of their body, is the center point (0,0,0).

When you pull the opponent forward only, you are increasing the Z value but the X and Y values remain at 0. Likewise, if I pull the opponent directly down (not towards me), the Y value decreases (because you’re going in the negative direction. The Z and X values for that pull are 0. Should I pull the opponent’s ear exactly on the horizontal plane, the X value increases, but Y and Z remain at 0.

How does “zero”-ing out one or more axis control the body? The body is amazing well equipped to deal with being pulled in “funny” directions. All joints and muscles can accommodate the pull and fold up. Also, it’s easy to pull someone in a “funny” direction because our bodies are full of joints and muscles. Should someone pull an opponent in only one axis, the body’s joints can not adjust to the steady pull.

The best example of this is the classic “mother ear pull”. The mother grabs the tender earlobes of her child and pulls them out of the store. She neither pulls down or to the side. These would be the X and Y-axis. It’s just a grab and she walks out of the store. Her walking is a pull in the Z.

**The Opposite for Joints**

Joint locks turn in two or more directions. Many of the locks require two joints to be placed in a 90° angle. For example, “nikajo” places the wrist at a 90° angle to the forearm. You also place the shoulder at a 90° to the body. For maximum effect, place the arm at 90° to the back of the uke. Other joint locks work in a similar fashion.

There’s a structural “law” in the application of the right angle to the skeletal system. The body has a difficult time recovering from such a position, unless it has been trained properly. The bones, ligaments and tendons are designed for “natural” positions. When they are placed in these specific angles, it looses its ability to move. Just Do It

Observe during your practice how this works. Fiddle with your partner to determine which fine application of the technique works best and most often. There is no substitute for working this out. You need to do and feel the subtle differences. This is how you make the art yours. You must practice, practice, and practice.