In the previous post I showed how to construct a triangle's circumcenter using taxicab geometry's version of a perpendicular line, the midset. There's another triangle center, but before we discuss it we have to explore angle measurement in taxicab geometry, which behaves very differently than it does in Euclidean geometry.
To illustrate, consider a circle with two 30° angles, one from 0° to 30°, the other from 30° to 60°:
Notice the arc lengths within the angles: the angle from 0° to 30° spans an arc that's almost 6 (Manhattan) units long, while the angle from 30° to 60° is just over 4 units long. The size of an angle in taxicab geometry, then, depends on how the angle is oriented to the axes. The biggest 1° angles are in the cardinal directions:
One degree of a Euclidean circle spans 1/360th of the circumference, but each of the above 1° angles spans about 1/230th of the taxicab circle's circumference, about one and a half times bigger.
The smallest 1° angles in taxicab geometry are perpendicular to the sides of a circle, at 45° angles to the axes:
These 1° angles span only about 1/460th of the circle, approximately three-quarters of a Euclidean degree.
Here are close-ups of the biggest and smallest 1° angles (on a finer grid):
The east-facing 1° angle spans an arc of almost 8 squares on the above grid, while the angle facing northeast only spans an arc of 4 squares, half as long.
These close-ups help us see what causes the difference. At a 45° angle from the center, a radius meets a circle at 90°, and the circle's perimeter traverses close to the shortest possible distance between the two radii. When a radius falls on the horizontal or vertical, however, it meets the circle at 45°, and the perimeter cuts at 45°, taking a much longer diagonal between two radii.
Thus, in taxicab geometry, if one degree means an angle that spans 1/360th of a circle's circumference, then 1° on the horizontal and vertical has to be narrower, and 1° on diagonals has to be wider. In the case of a unit taxicab circle, whose circumference is 8 units, 1° spans 8/360 units = 1/45 units = 0.02 units.