Taxicab2

Taxicab2 refers to the two-dimensional form of the Taxicab metric, commonly known as the Manhattan distance. It measures the distance between two points p = (x1, y1) and q = (x2, y2) in the plane as D2(p, q) = |x1 − x2| + |y1 − y2|. This metric embodies axis-aligned movement, in contrast to the straight-line distance of Euclidean geometry.

As a metric on the plane, Taxicab2 satisfies non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. The unit circle in this metric is a diamond shape (a square rotated 45 degrees), and shortest paths between points tend to follow axis-aligned segments. There are often multiple distinct shortest paths between two points, differing in the sequence of horizontal and vertical moves.

Taxicab2 is a special case of the L1 norm and generalizes to higher dimensions by summing absolute differences across all coordinates. Variants include weighted Manhattan distance, where different coordinates incur different costs: Dw(p, q) = w_x|dx| + w_y|dy|. In grid-based applications these distances are particularly convenient to compute and reason about.

Applications of Taxicab2 include robotics and autonomous navigation on grid maps, pathfinding in video games, image processing and computer vision tasks that use axis-aligned features, and clustering or similarity measures in settings where directional costs are anisotropic. The term is sometimes used to emphasize the two-dimensional, city-grid interpretation of the metric, and is related to broader discussions of taxicab geometry and Lp norms.

See also: Manhattan distance, taxicab geometry, L1 norm, distance metrics.