Circles of Apollonius

A circle of Apollonius is the set of points an equal ratio of distances from two foci. Stated algebraically,

As the name indicates, in Euclidean geometry this definition produces circles:

In taxicab geometry, it can produce a kite:

Or a trapezoid:

Given these two examples, and the fact that in Euclidean geometry circles of Apollonius are always circles, it's tempting to think that in taxicab geometry a circle of Apollonius is another way of defining a conic ellipse. But it's not. Here's a circle of Apollonius that isn't the cross section of a pyramid:

The jut from the lower corner of what would otherwise be a trapezoid is an interesting feature of circles of Apollonius in taxicab geometry. To understand why it appears, consider the case when the ratio of distances to each focus is 1. When that happens, the circle of Apollonius is equidistant from each focus, making the shape the set of points equidistant from both foci. In Euclidean geometry, that's a perpendicular bisector. Here are several circles between the same foci, with distance ratios approaching 1:

When the ratio is exactly 1, the circle's radius is infinite and the shape is a perpendicular bisector.

The same happens in taxicab geometry, where the set of points equidistant from two foci is a degenerate case of hyperbola. I typically call such a shape a "midset", since it's frequently not a single line nor perpendicular, more often a line with a pair of bends in it. It's also a circle of Apollonius with a distance ratio of 1:

The jut off one corner of some circles of Apollonius is how a growing shape transforms from a trapezoid into a midset.