Ideas in Geometry/Instructive examples/Lesson 6: Introduction to City Geometry

Introduction to City Geometry

In City Geometry, we have points and lines, just like in Euclidean Geometry. However, since we can only travel on city blocks, the distance between points is computed in a bit of a strange way. We don't measure distance as the crow flies. Instead we use the Taxicab distance:

Definition: Given two points A=(ax, ay) and B=(bx, by), we define the Taxicab distance as dT(A,B) = │ax-bx│+│ay-by│

Triangles: Triangle look the same in City Geometry as they do in Euclidean Geometry. Also, you measure the angles in exactly the same way. However, there is one minor hiccup. The lengths of the sides of each of these triangles are a little odd.

Circles: Circle is the collection of all points equidistant from a given point. So it City Geometry, we must conclude that a circle of radius 2 would look like a diamond two points away from the center.

Example: Will just bought himself a brand new gorilla suit. He wants to show it off at three parties this Saturday night. The parties are being held at his friends' house: The Antidisestablishment (A), Hausdorff (H), and the Wookie Loveshack (W). If she travels from party A to party H to Party W, how far does he travel this Saturday night?

Solution: We need compute

      dT (A,H) + dT (H,W)

Let's start by fixing a coordinate system and making A the origin. Then H is (2, -5) and W is (-10, -2). Then dT (A,H) = │0-2│+│0-(-5)│

        = 2+5
        = 7

and

dT (H,W) = │2-(-10)│+│-5-(-2)│

        = 12+3
        = 15

Will must trudge 7 + 15 = 22 blocks in his gorilla suit.

Midset Definition: Given two points A and B, their midset is the set of points that are equal distance away from both A and B.

-Michelle Hur