To go from a point to another we use a specific geometry : taxicab-geometry.

Distance is no more

A "straight line" between two points is meaningless :

there are several paths with same distance between two points A and B, that is several "taxi-lines".

How many taxi-lines from point A (0,0) to B (3,5) ? Solution

We can define a "taxi-circle" as the set of points at taxi-distance R from center O.

How many points on the taxi-circle with radius R = 12 ? Value of taxi-π ?
Solution

We want to go from A (0,0) to B (1,2).

The shortest path (taxi-line) has length x+y = 3.

As we have seen, there are several paths, and also we counted them.

Now the taxicab wants to go from A to B with d > x+y unit segments.

How many different paths ? for d = 5 ?
Solution

(forth and back on a same segment are allowed)

Taxi-ellipse : Set of points whose sum of taxi-distances to focuses F and F' is d.

Draw the taxi-ellipse with focuses (-3,0) and (0,3) with d = 12.
Solution

The taxi-ellipse with d = 13 ?
Solution