Two-dimensional case

In a standard two-dimensional phase space, all distances lie in a plane (Fig.1a). 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 (L in Fig. 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)0.5. If the x and y coordinates for point A are x₁, y₁ and those for point B are x₂, y₂ (Fig. 1a), then: length of side one = x coordinate for point B minus x coordinate for point A = x₂-x₁, and length of side two = y value for point B minus y value for point A = y₂-y₁.The desired distance (length L of the hypotenuse of the right triangle in Fig .1a) then is:

Figure .1: Distance between two points in phase space.