1. What is Manhattan Distance?

The Manhattan Distance, also known as Taxicab Distance or L1 Distance, is a metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. It's named after the grid layout of Manhattan streets where you can only move horizontally or vertically.

2. How Does the Calculator Work?

The calculator uses the Manhattan Distance formula:

Where:

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

Explanation: The formula calculates the distance by summing the absolute differences of the coordinates, representing the path a taxicab would take in a grid-like city.

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 any unit system (the result will be in the same units). The calculator works with both positive and negative coordinates.

5. Frequently Asked Questions (FAQ)

Q1: How is Manhattan Distance different from Euclidean Distance?
A: Euclidean Distance measures straight-line distance, while Manhattan Distance measures distance along axes at right angles (like city blocks).

Q2: Can Manhattan Distance be negative?
A: No, distance is always a non-negative value as it uses absolute differences.

Q3: Does the order of points matter in calculation?
A: No, the distance from A to B is the same as from B to A (commutative property).

Q4: Can this be extended to 3D space?
A: Yes, in 3D it would be \( D = |x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2| \).

Q5: Why is it called "Taxicab" distance?
A: Because it represents the distance a taxicab would travel in a city with a grid street system, where it can't cut diagonally through blocks.