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.
Understanding Geometry: Taxicab Geometry
The important step is to treat each coordinate change as a number of blocks, not as part of one diagonal move. A move from the point two, seven to the point negative three, four requires five blocks left and three blocks down. The directions do not cancel the distance traveled.
This is why absolute value is useful. It tells how far apart two coordinate values are without caring whether the motion is right or left, up or down. On graph paper, students can trace the route one block at a time and check that every counted block belongs to an allowed street.
There is often more than one shortest route. Suppose a trip needs four eastward blocks and three northward blocks. The driver can take the eastward blocks first, the northward blocks first, or mix them in many orders.
Every route that uses exactly those seven required blocks is equally short. In fact, there are thirty five different orders for these moves. This connects taxicab geometry to counting and probability.
Real streets may have road closures, one way rules, buildings, or crowded intersections. Then a route that would normally be shortest may not be available. The distance rule gives a lower limit, while the actual map decides whether that limit can be reached.
The shapes in this geometry show how strongly a distance rule affects a picture. Points at one fixed taxicab distance from a center form a diamond shaped boundary. Its sides run diagonally across the usual coordinate grid.
A region of points no more than a certain distance away fills the inside of that diamond. This idea helps with nearest location problems. For example, a delivery center can mark every grid location reachable within ten blocks.
Boundaries between areas served by different centers can look very different from the familiar curved boundaries based on straight line distance. The choice of geometry changes which store, hospital, or robot station counts as closest.
Taxicab distance appears in computer work whenever motion happens through separate steps. A pixel can move across rows and columns in an image. A warehouse robot may follow aisles.
A computer program can use taxicab distance as a quick estimate of how many grid moves remain before reaching a target. Students should pay attention to the rules before choosing a distance method. If diagonal movement is allowed, the answer can change.
If each street block has a different travel time, counting blocks alone is not enough. In that case, each block needs a weight based on time, cost, or energy. The main habit is to match the geometry to the path restrictions in the situation.
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.