## Cutting a corner

Why does strafe-running has the effect that it does? Well, by going
forward, you get a velocity of 50 units in the direction that you are
facing. By activating strafing and going left or right, you can get a
velocity of 50 units to one side. By doing both at the same time, you
move forward and to the side simultaneously, taking you along the
diagonal of a square. (This is where diagonal-running, the other
common term for this technique, has its origin.) Since the diagonal of
this square is longer than the side you'd move along by just going
forwards, you cover more ground in the same time.

The net result is that you move at 45^{o} to the angle you are
facing, but you also move faster. A little geometry shows that the
speed up factor you get should be equal to the square-root of two,
1.412... so the percentage increase is (sqrt(2) - 1) * 100%, or
approximately 41%.

That's the basic principle. Actually, things are a little more complicated than that, but
don't worry about it unless you are interested.