Distance

Distance measures how far apart two points are in a space. Its value depends on the space’s geometry and the rule used to measure separation.

Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. Distance becomes a metric when it obeys these four defining rules.
Euclidean distance: The square root of the sum of squared coordinate differences between two points in Euclidean space. This is the straight-line formula used in ordinary flat geometry.
Hamming distance: The number of positions at which two strings of equal length differ. It measures discrete disagreement instead of geometric straight-line separation.
Nearest neighbor search: The task of finding data points closest to a query under a chosen distance measure. Its results can change substantially when the distance rule changes.
Euclidean space: A geometric space modeled on ordinary flat geometry, with points represented by coordinate tuples. Its familiar distance rule measures straight-line separation between coordinate points.
Manhattan distance: The sum of absolute coordinate differences between two points on a grid. It measures axis-aligned travel rather than a straight-line segment.
Cosine similarity: A measure of the angle-based similarity between two nonzero vectors, independent of their magnitudes. It compares direction rather than how far apart vector endpoints are.
K-means clustering: An algorithm that partitions data into clusters by minimizing squared distances to cluster centers. Distance determines which observations belong to each cluster.
Norm: A function that assigns lengths to vectors and satisfies homogeneity, definiteness, and the triangle inequality. Norms turn vector differences into distances in many coordinate spaces.
Geodesic: A locally shortest path between points in a curved space or on a surface. On curved spaces, separation is measured along shortest paths constrained to the space.

Linked from 25 pages