Set of Points Equidistant from Two Points in Taxicab Geometry

In taxicab geometry, the usual Euclidean distance between points is replaced by the sum of the absolute differences of their coordinates. In symbols, if the two points are

(a,b)

and

(c,d)

, the distance between them is

|acb|+|b-d|

. The taxicab distance is also called Manhattan distance or rectilinear distance.

Drag the two red points (discretized only to help in checking) to see the set of points equidistant to them, which forms the "bisector line"

B

of the segment joining the points. If the two points are different, there are three possibilities for this "line":

1. When the line through the points has slope 0 or ∞,

B

coincides with the usual Euclidean bisector line.

2. When the line through the points has slope

±1

,

B

takes the shape of two infinite square regions (colored in blue) joined by a line segment.

3. Otherwise,

B

forms a zigzag line.