Euclidean distance

Euclidean distance 2d

Euclidean distance 3d 2 cropped

Minkowski distance examples

Euclidean distance is a measure of the true straight line distance between two points in Euclidean space. With this distance, Euclidean geometry becomes a metric space. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance.

Definition

The Euclidean distance between two points p and q is the length of the line segment connecting them (pq or qp). In the Cartesian coordinate system, if p = (p₁, p₂,..., p_n) and q = (q₁, q₂,..., q_n), then the distance (d) between p and q is given by:

d(p, q) = √[(p₁ - q₁)^2 + (p₂ - q₂)^2 + ... + (p_n - q_n)^2]

This formula is a direct application of the Pythagorean theorem.

Applications

Euclidean distance is used in many areas of science and engineering. It is fundamental in algorithms that solve problems involving distances, such as in machine learning for clustering and classification problems. In computer graphics, it is used to measure the distance between pixels or points in 3D modeling. In geography, it measures the actual distance between locations on a plane, assuming a flat earth, which is a good approximation for small distances.

Generalizations

The concept of Euclidean distance can be generalized to include points in any dimensional space. This generalization is straightforward and follows the same formula, adjusting for the number of dimensions. Furthermore, the concept of distance can be extended beyond Euclidean space to include other types of spaces, such as metric spaces, where the idea of distance can vary significantly from the Euclidean model.

See also

• Metric space
• Manhattan distance
• Chebyshev distance
• Minkowski distance

External links

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