Euclidean distance

Euclidean distance is the length of the straight line between two points in Euclidean space. In coordinates, it is the square root of the sum of squared coordinate differences.

Euclidean norm: The length of a vector in Euclidean space, computed as the square root of the sum of its squared coordinates. Euclidean distance is the Euclidean norm of the difference between the two points.
Point (geometry): An exact location in space with no size or extent. Euclidean distance measures separation between two such locations.
Manhattan distance: A distance measure equal to the sum of absolute coordinate differences. It follows axis-aligned paths rather than measuring the straight-line separation.
Nearest neighbor search: The task of finding data points closest to a query point under a chosen distance measure. Euclidean distance often defines closeness for numerical feature vectors.
Curse of dimensionality: The set of difficulties that arise as data dimension increases, including sparse samples and weakened distance contrasts. In high dimensions, Euclidean distances can become less discriminative between data points.
Pythagorean theorem: A theorem stating that a right triangle’s hypotenuse squared equals the sum of its legs’ squares. Repeated applications of this theorem yield the coordinate formula for distance.
Coordinate system: A scheme that assigns numerical coordinates to locations in a space. Coordinates make the distance calculable through differences in each dimension.
Chebyshev distance: A distance measure equal to the largest absolute coordinate difference. It measures the greatest single-coordinate displacement instead of combining all displacements quadratically.
k-nearest neighbors algorithm: A supervised learning method that predicts from the labels of nearby training examples. Its predictions depend on which examples are nearest under the selected distance.
Distance concentration: The tendency for distances among random points to become similar as dimension grows under many distributions. It can undermine nearest-neighbor methods based on Euclidean distance.

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