Definition
Taxicab geometry, also known as Manhattan geometry or ℓ₁ geometry, is a form of non‑Euclidean geometry in which the distance between two points is defined as the sum of the absolute differences of their Cartesian coordinates. Formally, for points $P=(x_1, y_1)$ and $Q=(x_2, y_2)$ in the plane, the taxicab distance (or Manhattan distance) is
$$
d_{\text{taxi}}(P,Q)=|x_2-x_1|+|y_2-y_1|.
$$
This distance function satisfies the axioms of a metric and therefore generates a metric space distinct from the Euclidean plane.
Overview
Taxicab geometry arises from the study of metric spaces equipped with the ℓ₁ norm. It models movement constrained to a rectilinear grid, such as the street layout of Manhattan, where travel occurs only along orthogonal axes. The geometry retains many concepts of classical geometry—lines, angles, circles, and polygons—but their properties differ markedly from their Euclidean counterparts. The subject is examined in textbooks on metric geometry, combinatorial geometry, and applied fields such as computer science, operations research, and urban planning.
Etymology / Origin
The term “taxicab” references the grid‑like street network of Manhattan, New York City, where taxicabs travel primarily along orthogonal avenues and streets. The alternative name “Manhattan distance” reflects the same origin. The concept of an ℓ₁ metric was formalized in the early 20th century within functional analysis and later popularized in geometric contexts by mathematicians such as Hermann Minkowski and later by educators illustrating alternative distance measures.
Characteristics
| Feature | Description |
|---|---|
| Metric | Defined by the ℓ₁ norm: $d_{\text{taxi}}(P,Q)=\sum_{i=1}^{n} |
| Unit “Circle” | The set of points at a fixed taxicab distance $r$ from a center forms a square rotated $45^\circ$ (a diamond) in the plane. |
| Geodesics | Shortest paths between two points are not unique; any monotone path that proceeds only toward the target along coordinate axes attains the minimal distance. |
| Angles | Conventional Euclidean angle measures lose direct geometric meaning; instead, notions such as “taxicab angle” are defined via alternative constructions (e.g., using the area of a sector). |
| Similarity & Scaling | Figures are similar under uniform scaling, but similarity transformations differ because dilation changes the shape of unit circles. |
| Coordinate Transformations | Rotations by angles other than multiples of $90^\circ$ do not preserve taxicab distance, unlike Euclidean rotations. |
| Applications | Widely used in city‑block routing, VLSI design, clustering algorithms (k‑medoids), and as a norm in optimization problems (e.g., linear programming). |
Related Topics
- Manhattan distance – the numerical measure of taxicab distance used in computer science and statistics.
- ℓₚ spaces – a family of normed vector spaces; taxicab geometry corresponds to the case $p=1$.
- Minkowski geometry – a broader class of normed geometries that includes taxicab geometry as a special case.
- Chebyshev distance (ℓ∞ norm) – another grid‑based metric where distance is the maximum coordinate difference.
- Non‑Euclidean geometry – the umbrella term for geometries that relax Euclidean axioms, of which taxicab geometry is one example.
- Metric space – the abstract setting in which distance functions like the taxicab metric are studied.
- Urban planning and routing algorithms – practical fields that frequently employ the taxicab metric for modeling movement on grid‑like networks.