ABSTRACT
In a standard two-dimensional phase space, all distances lie in a plane (Fig. 4.la). Ordinarily, you'll have the coordinates (x,y) of any two points for which you need the spanning distance. The distance from one point to the other is the length of the hypotenuse of a right triangle that we draw between the two points and lines parallel to the two axes. We can compute that hypotenuse length (Lin Fig. 4.1a) from the standard Pythagorean theorem. The theorem says that the square of the hypotenuse of a right triangle is the sum of (length of side one) 2 +(length of side two)2. Taking the square root of both sides of the equation gives the distance between the two points (the length of the hypotenuse L) as ([length of side one ]2 + [length of side two]2) 05 • If the x andy coordinates for point A are xi> y 1 and those for point Bare x2, y2 (Fig. 4.la), then:
(4.1)
As an example, say point A is at x1 = 1 and y 1 = 2 (written 1 ,2) and point B is at 8,6. The distance between the two points then is:
Distances and lines in space
Three-dimensional case
The Pythagorean theorem also is valid in three dimensions (Fig. 4.1b). It's written just as it is in the two-dimensional case but with the third variable (z) also included:
As in two dimensions, the components (here three) are distances measured along the three axes.
