Inversive geometry

Inversive geometry is a branch of mathematics that studies a specific type of transformation called inversion.

This process maps the Euclidean plane by moving points according to a reference circle. It is a highly effective tool for mathematicians. Many difficult geometric problems become much more tractable when an inversion is applied. This transformation has the unique ability to turn circles into lines and lines into circles. It also preserves the angles between crossing curves, even as the shapes themselves change.

To understand the mechanism, imagine a reference circle with a center $O$ and a radius $r$. When we invert a point $P$, we find its image, $P'$, on the same ray starting from $O$. The relationship is defined by the distance from the center. The product of the distances $OP$ and $OP'$ must equal $r^2$. This rule creates a specific movement pattern. A point inside the circle will always map to a point outside the circle. Conversely, any point outside the circle will map to a point inside it.

If a point is very close to the center $O$, its image will be very far away. If a point is far from the center, its image will be very close to the circle. Points located exactly on the reference circle are invariant, meaning they do not move at all.

To make this a complete mathematical function, we must account for the center $O$. Because we cannot divide by zero, the center $O$ is handled by introducing a single point at infinity. Inversion interchanges the center $O$ and this point at infinity. This makes the transformation an involution, which means applying the same inversion twice returns every point to its original position. This process can be visualized through various constructions. For a point outside the circle, one can use a midpoint and an intersecting circle to find the inverse.

Other methods, like Dutta's construction, work regardless of whether the point starts inside or outside the circle.

Inversive geometry was discovered by several mathematicians around the same time in the 19th century. Jakob Steiner indicated knowledge of the subject in 1824.

Adolphe Quetelet followed in 1825 by providing specific examples. The concept was also discovered independently by Giusto Bellavitis in 1836. Later, Stubbs and Ingram worked on it between 1842 and 1843. Finally, Lord Kelvin contributed to the field in 1845. These researchers helped establish how inversion affects different geometric sets.

The properties of inversion are quite diverse. A circle that passes through the center $O$ inverts into a straight line that does not pass through $O$. However, a line passing through $O$ inverts into itself.

If a circle does not pass through $O$, it inverts into another circle. Furthermore, if a circle or line is orthogonal to the reference circle, it remains unchanged by the inversion. Inversion also relates to the concept of poles and polars. A point $P$ can have a corresponding polar line that is perpendicular to the line containing the center and $P$.

Inversion can also be applied to three-dimensional space, known as sphere inversion. Instead of a reference circle, we use a reference sphere with radius $R$. The rules remain similar, but the shapes become more complex. A sphere that passes through the center of inversion will invert into a flat plane.

A sphere that does not pass through the center will invert into another sphere. This math can transform a cylinder, a cone, or a torus into a shape called a Dupin cyclide.

It can even transform a hyperboloid of one sheet into a different surface of revolution.

These mathematical ideas have practical applications in engineering and mapping. The Peaucellier–Lipkin linkage is a mechanical device that implements circle inversion. It provides an exact solution for converting circular motion into linear motion.

Inversion is also used in stereographic projection. This method maps a sphere onto a tangent plane. This is often done by treating the projection as an inversion of the sphere. From mechanical linkages to complex coordinate systems like 6-sphere coordinates, inversive geometry connects many different fields of study.