Taxicab

In big US towns, the streets build a rectangular network.
To go from a point to another we use a specific geometry : taxicab-geometry.
Distance is no more x² + y² but d = |x| + |y|.
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

Details

 

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