# Euclidean distance

In mathematics the Euclidean distance or Euclidean metric 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).

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## Definition

The Euclidean distance for two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is defined as

[itex]d(x,y):=\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}[itex]

## Two-dimensional distance

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

[itex]\sqrt{(px-qx)^2 + (py-qy)^2}[itex]

### 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 dydx, approximated 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%.

Missing image
Fasteuclid.png
image:fasteuclid.png

The following 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

[itex]\sqrt{(px-qx)^2 + (py-qy)^2 + (pz-qz)^2}[itex]

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