Euclidean distance

The Euclidean distance of two points x = (x₁,...,x_n) and y = (y₁,...,y_n) in Euclidean n-space is computed as

<math>\sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + \cdots + (x_n-y_n)^2} = \sqrt{\sum_{i=1}^n (x_i-y_i)^2}</math>

It is the "ordinary" distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. By using this formula as distance, Euclidean space becomes a metric space (even a Hilbert space).

Two-dimensional distance

For two 2D points P=[px,py] and Q=[qx,qy], the distance is computed as

<math>\sqrt{(px-qx)^2 + (py-qy)^2}</math>

Approximation

A fast approximation of 2D distance based on an octagonal boundary can be computed as follows. Let dx = |px-qx| (absolute value) and dy = |py-qy|. If dy≥dx, aproximated distance is 0.41dx+0.941246dy. (If dy<dx, swap these values.) The difference from the exact distance is between -6% and +3%; more than 85% of all possible differences are between -3% to +3%.

The following Waterloo Maple code implements this approximation and produces the plot on the right, with a true circle in black and the octagonal approximate boundary in red:

fasthypot :=
  unapply(piecewise(abs(dx)>abs(dy), 
                    abs(dx)*0.941246+abs(dy)*0.41,
                    abs(dy)*0.941246+abs(dx)*0.41),
          dx, dy):
hypot := unapply(sqrt(x^2+y^2), x, y):
plots[display](
  plots[implicitplot](fasthypot(x,y) > 1, 
                      x=-1.1..1.1, 
                      y=-1.1..1.1,
                      numpoints=4000),
  plottools[circle]([0,0], 1),
  scaling=constrained,thickness=2
);

Other approximations exist as well. They generally try to avoid the square root, which is an expensive operation in terms of processing time, and provide various error:speed ratio. Using the above notation, dx + dy - 2*min(dx,dy) yields error in interval 0% to 12%. (Attributed to Alan Paeth.)

Three-dimensional distance

For two 3D points P=[px,py,pz] and Q=[qx,qy,qz], the distance is computed as

<math>\sqrt{(px-qx)^2 + (py-qy)^2 + (pz-qz)^2}</math>