Manhattan Distance Calculator

Calculate the distance between two points using Taxicab Geometry (L1 Norm)

Enter the coordinates of two points to find the Manhattan distance. This metric is useful in grid-like paths, such as city blocks or chessboards.

Enter numbers separated by commas or spaces.

Enter numbers separated by commas or spaces.

Examples

Click on an example to load it into the calculator.

2D Urban Path

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Find the distance between two points in a 2D city grid.

P: [2, 3]

Q: [5, 7]

3D Space Navigation

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Calculate the distance in a 3-dimensional coordinate system.

P: [10, -5, 8]

Q: [4, 2, 12]

Data Science Feature Vector

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Measure the distance between two 4D feature vectors in a dataset.

P: [1, 0, 1, 0]

Q: [0, 1, 0, 1]

Simple 1D Distance

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Calculate the distance between two points on a number line.

P: [-15]

Q: [25]

Other Titles
Understanding Manhattan Distance: A Comprehensive Guide
Explore the concept of Manhattan Distance, its formula, applications, and how it differs from Euclidean distance.

What is Manhattan Distance?

  • Also known as Taxicab Distance or L1 Norm.
  • Measures distance on a grid, moving only horizontally and vertically.
  • Represents the sum of the absolute differences of the coordinates.
Manhattan Distance, also known as L1 Distance, Taxicab Geometry, or City Block Distance, is a way of measuring distance between two points in a space. Unlike the more common Euclidean distance (a straight line), the Manhattan distance is the distance a taxi would travel in a city laid out in a grid, moving only along horizontal and vertical streets.
The name comes from the grid-like street layout of Manhattan in New York City. Imagine you want to get from point P to point Q. You can't go through buildings; you must follow the streets. The total distance you travel—the sum of the horizontal blocks and vertical blocks—is the Manhattan distance.
The Formula
For two points P = (p₁, p₂, ..., pₙ) and Q = (q₁, q₂, ..., qₙ) in an n-dimensional space, the Manhattan distance (d₁) is calculated as:
d₁ = Σ |pᵢ - qᵢ| from i=1 to n
In a 2D plane with points P(x₁, y₁) and Q(x₂, y₂), the formula simplifies to: d₁ = |x₁ - x₂| + |y₁ - y₂|

Basic Examples

  • For P(1, 2) and Q(4, 6), distance = |1-4| + |2-6| = 3 + 4 = 7.
  • On a chessboard, the minimum number of moves for a king to go from one square to another is the Manhattan distance.

Step-by-Step Guide to Using the Calculator

  • Enter coordinates for two points.
  • Ensure both points have the same dimension.
  • Interpret the calculated distance and steps.
Our calculator simplifies finding the Manhattan distance. Follow these steps for an accurate calculation.
Input Guidelines
1. Point 1 Coordinates: In the first input field, enter the coordinates of your starting point. You can use commas (e.g., 10, 20, 5) or spaces (e.g., 10 20 5) to separate the numbers.
2. Point 2 Coordinates: In the second input field, enter the coordinates of your ending point, using the same format.
3. Dimensional Consistency: Crucially, both points must have the same number of coordinates (dimensions). You cannot calculate the distance between a 2D point and a 3D point.
Calculation and Results
  • Click the 'Calculate Distance' button to perform the calculation.
  • The 'Manhattan Distance' will be displayed, showing the final numerical result.
  • The 'Calculation Steps' section provides a detailed breakdown, showing the absolute difference for each dimension and how they are summed together, making the process easy to understand and verify.

Practical Usage Examples

  • Input P: `1, 2`, Q: `3, 4` -> Steps: |1-3| + |2-4| = 2 + 2 = 4.
  • Input P: `-5, 8`, Q: `5, -2` -> Steps: |-5-5| + |8-(-2)| = 10 + 10 = 20.

Real-World Applications of Manhattan Distance

  • Pathfinding in games and robotics.
  • Feature distance in machine learning.
  • Image analysis and computer vision.
Manhattan distance is not just an abstract concept; it has practical applications in various fields.
Computer Science and AI
  • Robotics & Game Development: It's used for pathfinding on grids. Algorithms like A* can use Manhattan distance as a heuristic to estimate the cost to reach a destination in games with grid-based movement.
  • Machine Learning: In classification and clustering, Manhattan distance can be used to measure the dissimilarity between data points, especially in high-dimensional spaces where Euclidean distance can be counter-intuitive (curse of dimensionality).
Image Processing
In computer vision, it's used to compare images. By treating pixel values as coordinates in a high-dimensional space, Manhattan distance can measure the difference between two images.
Urban Planning & Logistics
Delivery services and urban planners use this metric to estimate travel times and distances in cities with a grid-like street network, optimizing routes and service areas.

Industry Applications

  • A chess AI calculating the minimum moves for a rook.
  • A K-Nearest Neighbors (KNN) algorithm using L1 norm for feature comparison.
  • An autonomous warehouse robot navigating aisles to pick up items.

Manhattan vs. Euclidean Distance

  • Euclidean is the 'as-the-crow-flies' distance.
  • Manhattan is the grid-based or path-based distance.
  • The choice depends on the problem's constraints.
Understanding the difference between Manhattan and Euclidean distance is key to applying them correctly.
The Core Difference
  • Euclidean Distance (L2 Norm): This is the straight-line distance between two points. It's the shortest possible path, calculated using the Pythagorean theorem: sqrt(Σ(pᵢ - qᵢ)²).
  • Manhattan Distance (L1 Norm): This is the distance along axes at right angles. It's always greater than or equal to the Euclidean distance.
When to Use Which?
  • Use Euclidean Distance when movement is unrestricted and can happen in any direction. For example, calculating the distance for an airplane's flight path.
  • Use Manhattan Distance when movement is constrained to a grid or orthogonal directions. This is common in urban environments, circuit board design, or bioinformatics for sequence alignment.

Comparison Examples

  • P(0,0) to Q(3,4): Euclidean = sqrt(3²+4²) = 5. Manhattan = |3-0|+|4-0| = 7.
  • For points on the same axis, like P(0,0) and Q(5,0), both distances are the same (5).

Mathematical Derivation and Properties

  • Derived from the concept of vector norms.
  • Satisfies the properties of a metric: non-negativity, identity, symmetry, and triangle inequality.
  • Geometrically, the 'circles' are squares.
Manhattan distance is part of a broader family of distance metrics derived from vector norms.
Vector Norms
In mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space. The Manhattan distance is derived from the L1 norm of the difference between two vectors.
Properties of a Metric
Manhattan distance is a true metric, meaning it satisfies four key conditions:
1. Non-negativity: d(P, Q) ≥ 0
2. Identity of indiscernibles: d(P, Q) = 0 if and only if P = Q
3. Symmetry: d(P, Q) = d(Q, P)
4. Triangle Inequality: d(P, R) ≤ d(P, Q) + d(Q, R)
Geometric Interpretation
A fascinating aspect is the shape of a 'circle' in Manhattan geometry. A circle is defined as the set of points equidistant from a center point. While in Euclidean geometry this is a familiar round circle, in Manhattan geometry, it's a square rotated by 45 degrees.

Mathematical Insight

  • The set of points with a Manhattan distance of 3 from (0,0) includes (3,0), (2,1), (1,2), (0,3), (-1,2), etc., forming a diamond shape (a rotated square).
  • The triangle inequality ensures that going directly from P to R is never longer than going via an intermediate point Q.