#inversive

Inversive Geometry is a small subset of the mathematical subject known as geometry that we were taught at some point in our lives. This topic of geometry isn't usually talked about until higher levels of learning, but the fundamentals can be taught with only prior knowledge of basic geometry.

Inversive geometry works by having a starting point, usually called the inversion center, and an inversion circle around the inversion center with a radius \( k \) called the inversion radius. From these geometric constructions, we can find anything from inverse point to an inverse curve which is a locus of inverse points. It's interesting to note that the inverse curve of an inverse curve is the original curve--very similar to how an inverse function or a reciprocal would work. Many inverse curves produced from well-known curves also produce other well-known curves.  For example, the inverse curve of a parabola is a cardioid. An example of this will be shown at the end.

The most primitive example of inversive geometry is inverting a single point. Let's call this point P. To start off, we create the inversion center O and inversion circle \( O \). Then, a length is drawn from the inversion center O to the point P. A circle with diameter OP is drawn which will intersect the inversion circle \( O \) twice. The intersections of these two circles will be called N and N'. (A special property of these two points is that the line formed between P and either N or N' will always be perpendicular to the line between O and either N or N'.) A line between these two points N and N' is constructed. The inverse point P' is the result of the intersection between this line and the original line between O and P. This can be seen in the image below for clarity.

[Image taken from Wikipedia Entry on Inversive Geometry]

As a result of this procedure, points that are on the inversion circle map to the exact same points when inverted.

If this procedure is repeated for all the points within the domain of a plane curve, then we end up with the inverse curve as the locus of all those points. For an example of this, we look towards the parabola. It is easiest to visualize this in polar coordinates due to a very simple formula for finding the inverse curve of a polar equation with respect to the unit circle:

$$ r_{inv} = \frac{k^2}{r_{orig}} $$

For this formula, \( k \) represents the radius of the inversion circle. The polar equation for a parabola is \( r = \frac{l}{1- \cos (\theta)} \) where \( l \) is half of the latus rectum. To find the inverse curve of this parabola with respect to the unit circle ( \( k = 1 \) ), we use the formula and find that the equation for the inverse curve is \( r = \frac{1}{l}(1 - \cos (\theta)) \), which is the polar equation for a cardioid scaled by a factor of \( l \). This can be seen in the image below. Notice how the points on the inversion circle are where the two inverse curves intersect each other.

[Image taken from Wikimedia Commons]