📚 65th St NYC: Taxicab Geometry and the Manhattan Metric | 纽约65街：出租车几何与曼哈顿度量
Imagine you are standing at the intersection of 65th Street and 3rd Avenue in Manhattan, New York City. You need to walk to a café at 72nd Street and Lexington Avenue. Since the city is laid out in a rectangular grid of streets and avenues, you cannot cut diagonally through buildings; you must travel along the streets (east–west) and avenues (north–south). The shortest walking distance is not the straight-line Euclidean distance one would measure on a map, but the sum of the horizontal and vertical blocks travelled. This is the essence of the Manhattan distance, the foundational concept of taxicab geometry.
想象你正站在纽约曼哈顿第65街与第三大道的交叉口，需要步行前往第72街与列克星敦大道的一家咖啡馆。由于城市布局呈矩形网格，街道和大道纵横交错，你无法斜穿建筑群，只能沿着街道（东西向）和大道（南北向）行走。实际的步行距离并非地图上测量出的直线欧几里得距离，而是横向与纵向所经过街区数的总和。这就是曼哈顿距离的精髓，也是出租车几何学的基础概念。
1. Manhattan Streets and Coordinate Geometry | 曼哈顿街道与坐标几何
New York City's borough of Manhattan, especially above 14th Street, is famously organised as a grid. Avenues run north–south (e.g. 1st, 2nd, 3rd Avenue, …) and streets run east–west (e.g. 42nd St, 65th St, 72nd St). This makes it an ideal model for a Cartesian coordinate plane. We can assign an x-coordinate to the avenue number and a y-coordinate to the street number. For instance, let the intersection of 3rd Avenue and 65th Street be point A with coordinates (3, 65). If your destination is 5th Avenue and 72nd Street, point B has coordinates (5, 72). The change in x is 2 avenues, and the change in y is 7 streets.
纽约市曼哈顿区，尤其是14街以北，以著名的网格状布局规划。大道呈南北走向（如第一大道、第二大道、第三大道……），街道呈东西走向（如第42街、第65街、第72街）。这为建立笛卡儿坐标平面提供了绝佳模型。我们可以将大道编号赋予 x 坐标，街道编号赋予 y 坐标。例如，设第三大道与第65街的交叉口为点 A，坐标为 (3, 65)。若目的地是第五大道与第72街，点 B 的坐标为 (5, 72)。x 的变化量为 2 条大道，y 的变化量为 7 条街道。
2. The Manhattan Distance Formula | 曼哈顿距离公式
₁)²). However, in a grid-based city like Manhattan, the distance a taxi must travel is the sum of the absolute differences in the x and y coordinates. This is called the Manhattan distance.
在标准的欧几里得几何中，两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离为 dₑ = √((x₂ − x₁)² + (y₂ − y₁)²)。然而，在曼哈顿这样的网格状城市中，出租车行驶的距离必须是 x 坐标差值的绝对值与 y 坐标差值的绝对值之和。这就是曼哈顿距离。
dₘ = |x₂ − x₁| + |y₂ − y₁|
For our example from (3, 65) to (5, 72), dₘ = |5 − 3| + |72 − 65| = 2 + 7 = 9 blocks.
对于我们举例的从 (3, 65) 到 (5, 72)，dₘ = |5 − 3| + |72 − 65| = 2 + 7 = 9 个街区。
3. Euclidean vs Manhattan: A Comparison | 欧几里得距离与曼哈顿距离对比
For the same two points, the Euclidean distance is dₑ = √2² + 7² = √4 + 49 = √53 tan distance (9) is always greater than or equal to the Euclidean distance, with equality only when the points lie on a horizontal or vertical line. The difference highlights the 'detour' nt. If a bird could fly directly, it would cover 7.28 units, but a pedestrian must walk 9 blocks. This simple relationship forms the basis of taxicab geometry, a non-Euclidean metric that satisfies all conditions of a distance function.
对于同样的两个点，欧几里得距离为 dₑ = √2² + 7² = √4 + 49 = √53 ≈ 7.28 个街区。请留意，曼哈顿距离 (9) 总是大于或等于欧几里得距离，只有当两点处在同一水平线或垂直线时两者才会相等。这一差异突显了由于网格限制而必须绕行的特征。如果鸟儿可以径直飞过，其飞行距离为 7.28，但行人必须步行 9 个街区。这一简单关系构成了出租车几何学的基础，这是一种满足距离函数所有条件的非欧几里得度量。
4. Multiple Shortest Paths and Combinatorics | 最短路径的数量与组合数学
ue blocks east and 7 street blocks north, the path can be represented as a sequence of 2 E's and 7 N' er of distinct shortest paths is given by the binomial coefficient C(9, 2) = C(9, 7) = 36.
在网格中，最短路径并非唯一；多种不同的移动序列都可以达到相同的曼哈顿距离。对于需要向东走过 2 个大道街区、向北走过 7 个街道街区的行程，路径可以表示为 2 个“东”和 7 个“北”的序列。由恰好 2 次向东和 7 次向北构成的 9 步任意排列，都属于有效的最短路径。不同的最短路径总数由二项式系数 C(9, 2) = C(9, 7) = 36 给出。
Number of paths = C(a + b, a) where Δx = a, Δy = b
More generally, if the horizontal displacement is a and vertical displacement is b, the number of shortest paths is C(a + b, a). This connects directly to Pascal's triangle and combinatorial counting.
更一般地，若水平位移为 a，垂直位移为 b，则最短路径的数量为 C(a + b, a)。这与帕斯卡三角及组合计数直接相关。
5. A Circle in Taxicab Geometry | 出租车几何中的圆
ed by 45°. For example, a taxicab circle of radius 3 centred at the origin consists of all points (x, y) such that |x| + |y| = 3.
在欧几里得几何中，圆是到中心点距离为常数 r 的所有点的集合。出人意料的是，在出租车几何中，满足 dₘ = r 的点集看起来像一个旋转了 45° 的正方形。例如，以原点为中心、半径为 3 的出租车圆包含所有满足 |x| + |y| = 3 的点 (x, y)。
|x| + |y| = r
This produces a diamond shape with vertices at (r, 0), (0, r), (−r, 0) and (0, −r). The circumference (total Manhattan length along the boundary) is 8r, and the area (counting unit squares) is 2r² — remarkably different from Euclidean values of 2πr and πr².
该图形呈现为菱形，顶点在 (r, 0)、(0, r)、(−r, 0) 和 (0, −r)。其周长（沿边界的曼哈顿距离总和）为 8r，其面积（计数单位方块）为 2r²——与欧几里得圆的周长 2πr 和面积 πr² 截然不同。
6. Properties of the Taxicab Metric | 出租车度量的性质
The Manhattan distance
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