1. What is Manhattan Distance?

The Manhattan Distance, also known as Taxicab Distance or L1 Distance, is a metric between two points in a grid-based system where movement is restricted to horizontal and vertical paths only. It's the sum of the absolute differences of their Cartesian coordinates.

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 formula calculates the distance between two points when you can only move along grid lines (like a taxi moving through city blocks).

3. Applications of Manhattan Distance

Details: Manhattan Distance is widely used in computer science (pathfinding algorithms), data analysis (clustering), image processing, and any application where movement is constrained to grid-like paths.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: How is Manhattan Distance different from Euclidean Distance?
A: Manhattan Distance sums the absolute differences of coordinates, while Euclidean Distance is the straight-line distance (square root of sum of squared differences).

Q2: Why is it called "Manhattan" Distance?
A: It's named after the grid-like street layout of Manhattan, where you can't cut diagonally through buildings.

Q3: Can this be extended to higher dimensions?
A: Yes, in n-dimensional space, it's the sum of absolute differences along each dimension: \( \sum_{i=1}^n |x_i - y_i| \).

Q4: What's the maximum Manhattan Distance between two points?
A: In a bounded space, it depends on the grid size. For an n×n grid, the maximum is 2n (from opposite corners).

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