# euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between 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). Older literature refers to this metric as Pythagorean metric. The technique has been rediscovered numerous times throughout history, as it is a logical extension of the pythagorean theorem.

## Definition

The Euclidean distance between points and , in Euclidean n-space, is defined as:

## One-dimensional distance

For two 1D points, and , the distance is computed as:

The absolute value signs are used, since distance is normally considered to be an unsigned scalar value.

## Two-dimensional distance

For two 2D points, and , the distance is computed as:

Alternatively, expressed in circular coordinates (also known as polar coordinates), using and , the distance can be computed as:

### 2D approximations for computer applications

A fast approximation of 2D distance based on an octagonal boundary can be computed as follows. Let (absolute value) and . If , approximated distance is . (If , 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 Maple code implements this approximation:
```>
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 − (1/2)×min(dx,dy) yields error in interval 0% to 12% (attributed to Alan Paeth). A better approximation in term of RMS error is: dx + dy - (5/8)×min(dx,dy) and yields error in interval −3% to 7%.

Also note that when comparing distances (for which is greatest, not for the actual difference), it isn't necessary to take the square root at all. If distance is greater than distance , then will also be greater than . Or, when checking to see if distance is greater than , that is the same as comparing with or , etc. An example of the first case might be when trying to determine which nearest grid point an arbitrary point should "snap to" in a 2D CAD/CAM system. This isn't really an approximation, however, as the results are exact.

## Three-dimensional distance

For two 3D points, and , the distance is computed as

### 3D approximations for computer applications

As noted in the 2D approximation section, when comparing distances (for which is greatest, not for the actual difference), it isn't necessary to take the square root at all. If distance is greater than distance , then will also be greater than . An example is when searching for the minimum distance between two surfaces in 3D space, using a 3D CAD/CAM system. One way to start would be to build a point grid on each surface, and compare the distance of every grid point on the first surface with every grid point on the second surface. It isn't necessary to know the actual distances, but only which distance is the least. Once the closest two points are located, a much smaller point grid could be created around those closest points on each surface, and the process repeated. After several iterations, the closest two points could then be fully evaluated, including the square root, to give an excellent approximation of the minimum distance between the two surfaces. Thus, the square root only needs to be taken once, instead of thousands (or even millions) of times.

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
Distance is a numerical description of how far apart objects are at any given moment in time. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over").
In mathematics, the Pythagorean theorem (AmE) or Pythagoras' theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.
Hilbert space, named after the David Hilbert, generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed in terms of
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.
Maple is a general-purpose commercial mathematics software package. It was first developed in 1980 by the Symbolic Computation Group at the University of Waterloo in Waterloo, Ontario, Canada.

Since 1988, it has been developed and sold commercially by Waterloo Maple Inc.
Expand article, tagged for improvement in January 2007,date January 2007

CAD CAM is an abbrieviation of computer-aided design and computer aided manufacturing. CAD

Computer Aided Design CAD this system uses something called TTUA is the use of a wide range of tools that
Expand article, tagged for improvement in January 2007,date January 2007

CAD CAM is an abbrieviation of computer-aided design and computer aided manufacturing. CAD

Computer Aided Design CAD this system uses something called TTUA is the use of a wide range of tools that