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Taxicab Geometry > Midset of two points

In Euclidean geometry, the midset of two points is the perpendicular bisector of the line segment connecting the points. If we mimic the circle expansion strategy employed in the discussion about the distance from a point to a line, we can easily see this (Figure 1).

FIGURE 1: Forming the Euclidean midset of two points using expanding circles.

When this approach is employed in taxicab geometry, four distinct cases appear based on how expanding taxicab circles intersect. The first case shown in Figure 2 might easily lull you into thinking life has not changed much from Euclidean geometry. When the line connecting the points is horizontal or vertical, expanding taxicab circles about the points initially meet at a single point - the midpoint of the line segment connecting the points. As the circles continue to expand, the circles meet at exactly two points. This produces the same line as the Euclidean midset of the points - the Euclidean perpendicular bisector of the line segment connecting the points.

FIGURE 2: The taxicab midset of two points lying along a horizontal line.

If the line connecting the points is not horizontal or vertical, things begin to change. The expanding circles about the points initially meet in a line segment that passes through the midpoint of the line segment connecting the points (Figure 3). As the circles continue to expand, they meet at exactly two points. The expanding circles create two rays moving away from the original points.

FIGURE 3: The taxicab midset of two points lying along a gradual line.

Note that this case actually breaks down into two cases. The rays formed by the intersections of the expanding circles have a different orientation depending on the slope of the line segment connecting the original points. Figure 4 shows a line segment with a slope of magnitude greater than 1 whereas Figure 3 illustrated a line segment with slope of magnitude less than 1.

FIGURE 4: The taxicab midset of two points lying along a steep line.

This leaves only the case where the slope of the line connecting the points has magnitude 1 (Figure 5). In this case the expanding circles also initially meet in a line segment. But, as the expansion continues, the circles meet in line segments instead of points. This creates a midset consisting of two regions of points joined by a diagonal line.

FIGURE 5: The taxicab midset of two points lying along a diagonal line.

Only when the line segment connecting the points is horizontal or vertical is the midset of the two points a simple line like the Euclidean midset of two points. In all other cases the midset is at best a piecewise-defined line and at worst not a line at all! As with Euclidean geometry, the midset always includes the midpoint of the line segment connecting the points.

References
[1] Krause, Eugene F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry, 1975.

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