Inversive geometry

In geometry, inversive geometry is the study of inversion. Inversion is a special way to change shapes on a flat surface. It takes circles or straight lines and turns them into other circles or lines. It keeps the angles between them the same. This makes it easier to solve many hard geometry problems.

The idea of inversion was discovered by several people around the same time. Steiner found it in 1824, Quetelet in 1825, Bellavitis in 1836, Stubbs and Ingram between 1842 and 1843, and Kelvin in 1845. Inversion can also be used in spaces with more than two dimensions.

Inversion in a circle

The inverse of a point in geometry is a special way to find a new point related to a given circle. Imagine you have a point P and a circle with center O and radius r. The inverse point P' lies on the same line from O through P, but its distance from O is such that the product of the distances OP and OP' equals r squared. This means points closer to the center move farther away, and points farther away move closer.

Inversion has many interesting properties. For example, a line not passing through the center of the circle becomes a circle that does pass through the center, and a circle passing through the center becomes a line not passing through the center. Inversion also keeps angles the same size but can change their direction. These properties make inversion useful for solving many geometry problems more easily.

In three dimensions

In three dimensions, inversion works like it does in two dimensions but uses spheres instead of circles. When you invert a point in 3D space using a special sphere, the point moves along a straight line from the center of the sphere to a new spot. This keeps angles the same, just like in two dimensions.

In three dimensions, spheres turn into other spheres. If a sphere goes through the center of the special sphere, it changes into a flat plane. Planes that go through the center also become spheres. This idea helps make many hard geometry problems easier to solve.

Axiomatics and generalization

One of the first people to study inversive geometry was Mario Pieri in 1911 and 1912. Edward Kasner wrote his thesis on the "Invariant theory" of the inversion group.

Later, the rules of inversive geometry were seen as a special type of structure. In this structure, certain shapes are called "blocks." In incidence geometry, any flat area with one extra point forms something called a Möbius plane, also known as an inversive plane. There are versions of these planes that have a limited number of points and others that go on forever. A way to model the Möbius plane using normal flat shapes is called the Riemann sphere.

Invariant

The cross-ratio between four points stays the same even after an inversion. This helps make solving geometry problems easier. When you flip or turn shapes using inversion, some measurements change, but the cross-ratio does not.

Relation to Erlangen program

The transformation called inversion in a circle was invented by L. I. Magnus in 1831. Since then, this mapping has become important in higher mathematics.

Through using circle inversion, students of transformation geometry can learn about the importance of Felix Klein's Erlangen program. This idea grew from models of hyperbolic geometry.

Combining two inversions in circles that share the same center results in a similarity or homothetic transformation, called dilation. This dilation is set by the ratio of the radii of the two circles.

Inversive geometry also links to complex numbers and the Möbius group. This group includes changes like moving and turning. Adding reciprocation—based on circle inversion—helps make the special features of Möbius geometry. However, inversive geometry is broader. It also includes basic circle inversion and conjugation mappings.

In higher dimensions

In higher dimensions, inversive geometry uses a special change called inversion. This change moves points in space using a sphere. It flips points based on how far they are from the center of the sphere, creating new shapes and movements.

This idea helps us understand how objects can stretch, slide, or turn in space. When we mix these movements, we get special maps that keep angles the same. These maps are important for studying shapes and spaces.

Anticonformal mapping property

The circle inversion map is called "anticonformal" because it keeps the sizes of angles between lines and curves the same but flips which way they face. This is different from a "conformal" map, which keeps both the size and direction of angles.

In math, this means the special grid of numbers (called a Jacobian) for this map is a simple stretch or shrink of a mirror flip. For complex numbers, the basic circle inversion map uses the mirror image of the normal inverse map, making it anticonformal instead of conformal.

Hyperbolic geometry

The (n − 1)-sphere can be described by a special kind of equation. When we use a process called inversion, some of these spheres stay exactly the same. These special spheres are important in a model of hyperbolic geometry called the Poincaré disk model.

Inversion in the unit sphere keeps certain spheres unchanged and swaps points inside the sphere with points outside. This helps us understand how angles and shapes behave in hyperbolic geometry.