1. What is Manhattan Distance?

The Manhattan Distance, also known as Taxicab Distance or L₁ Distance, is a measure of the distance between two points in a grid-based system where movement is restricted to horizontal and vertical paths only, much like navigating the streets of Manhattan.

2. How Does the Calculator Work?

The calculator uses the Manhattan Distance formula:

Where:

• \( (x_1, y_1) \) — Coordinates of the first point
• \( (x_2, y_2) \) — Coordinates of the second point
• \( | \cdot | \) — Absolute value function

Explanation: The distance is calculated by summing the absolute differences of their Cartesian coordinates.

3. Applications of Manhattan Distance

Details: Manhattan Distance is widely used in computer science (pathfinding algorithms), data analysis (clustering), image processing, and urban planning.

4. Using the Calculator

Tips: Enter the coordinates of two points in 2D space. The calculator will compute the Manhattan Distance between them. All values can be positive or negative real numbers.

5. Frequently Asked Questions (FAQ)

Q1: How is Manhattan Distance different from Euclidean Distance?
A: Manhattan Distance sums absolute differences (grid path), while Euclidean Distance uses straight-line distance (Pythagorean theorem).

Q2: Can this be extended to higher dimensions?
A: Yes, for n-dimensional points, sum absolute differences across all dimensions: \( \sum_{i=1}^n |x_i - y_i| \).

Q3: Why is it called "Manhattan" Distance?
A: It resembles the path a taxi would take in Manhattan's grid-like street layout, unable to cut diagonally through buildings.

Q4: What's the maximum Manhattan Distance between two points?
A: In a bounded area, it depends on the grid size. For a square of side length L, maximum distance is 2L (between opposite corners).

Q5: When should I use Manhattan vs Euclidean Distance?
A: Use Manhattan when movement is grid-constrained (like chess), Euclidean when straight-line distance is possible (like flying).