Taxicab geometry measures distance the way a taxi drives through a city with square blocks. Instead of moving along a straight diagonal, travel is limited to horizontal and vertical streets on a grid. This matters because many real paths, such as city driving, robot motion, circuit layouts, and warehouse navigation, are closer to grid movement than straight-line movement.
The main idea is that the shortest allowed route may look like a stair-step path instead of a straight segment.
In the coordinate plane, the Euclidean distance between A(x1, y1) and B(x2, y2) is found with the Pythagorean theorem, while taxicab distance adds the horizontal and vertical changes. The taxicab distance is dT = |x2 - x1| + |y2 - y1|. Because of this rule, all shortest taxicab paths between two points have the same total length if they use only the needed horizontal and vertical moves.
A taxicab circle looks like a tilted square because every point on it has the same total horizontal plus vertical distance from the center.
Key Facts
- Euclidean distance: dE = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- Taxicab distance: dT = |x2 - x1| + |y2 - y1|.
- Taxicab distance is also called Manhattan distance or city-block distance.
- For A(1, 2) and B(6, 5), dT = |6 - 1| + |5 - 2| = 8.
- A taxicab circle centered at (h, k) with radius r satisfies |x - h| + |y - k| = r.
- When both horizontal and vertical changes are nonzero, taxicab distance is greater than Euclidean distance for the same two points.
Vocabulary
- Taxicab geometry
- A geometry where distance is measured by adding horizontal and vertical movement along a grid.
- Euclidean distance
- The straight-line distance between two points in the usual plane.
- Manhattan distance
- Another name for taxicab distance, based on travel along city blocks.
- Metric
- A rule for measuring the distance between points.
- Taxicab circle
- The set of all points that are the same taxicab distance from a center point.
Common Mistakes to Avoid
- Using the Pythagorean theorem for taxicab distance is wrong because taxicab paths cannot use diagonal shortcuts.
- Forgetting absolute values in dT = |x2 - x1| + |y2 - y1| is wrong because distance cannot be negative.
- Thinking there is only one shortest taxicab path is wrong because many different stair-step routes can have the same total horizontal and vertical length.
- Drawing a taxicab circle as a round circle is wrong because points at a fixed taxicab distance form a diamond-shaped square on the grid.
Practice Questions
- 1 Find the Euclidean distance and taxicab distance between A(2, 1) and B(8, 5).
- 2 A taxi travels from (3, 7) to (10, 2) on a square street grid. What is the taxicab distance?
- 3 Explain why a taxicab circle centered at the origin with radius 4 has corners at (4, 0), (0, 4), (-4, 0), and (0, -4) instead of forming a round shape.