exupero's blog

Parallel line construction in taxicab geometry

In the previous post I glossed over one step in bisecting taxicab angles: the construction of parallel lines. We can make such lines in either of two ways: using Euclidean compass and straightedge construction, or using taxicab circles.

Parallel lines in taxicab geometry aren't any different than in Euclidean geometry, only how the distance between the lines is measured. Thus, if we have a point the right distance from the line, we can use any regular compass and straightedge construction that produces a parallel line through a point. For lines angled closer to the horizontal axis, pick a point the desired distance above or below the line; for lines closer to the vertical, pick a point left or right of the line.

If we use taxicab circles, however, there's a shortcut. We only need to draw two circles of the same radius centered on the line, like this:

We can construct parallel lines by placing a straightedge against the corners of the circles:

Technically this construction works with Euclidean circles too, but it has the disadvantage that the straightedge can't be placed on any established points and its placement tangent to the circles has to be eyeballed. A book I have on compass and straightedge construction calls it a "cheat". In taxicab geometry, however, the corners of the circles serve as established points.

The trick now is how to reliably draw a taxicab circle.