Ideal point

An ideal point is a special idea used in a type of math called hyperbolic geometry. In this geometry, which is different from the flat geometry we see around us, ideal points exist outside the main space we study. They help us understand how lines behave when they go far away.

Imagine you have a line and a point not on that line. If you draw special lines called limiting parallels through that point, they seem to point toward certain places far away. These places are called ideal points. They are not part of the space itself, but they help us describe how lines meet at the edges of this geometry.

Ideal points together make up something called the Cayley absolute, which is like a boundary for hyperbolic space. For example, in models like the Poincaré disk model or the Klein disk model, a circle around the edge represents these ideal points. These points help mathematicians study shapes and angles in curved spaces in interesting ways.

Properties

In hyperbolic geometry, an ideal point is a special point that exists outside the main space we study. The distance from any ideal point to other points or even to other ideal points is always infinite. These points act as centers for shapes called horocycles and horoballs. When two horocycles share the same ideal point as their center, they are known as concentric.

Polygons with ideal vertices

Main article: Ideal triangle

When all the corners of a triangle are ideal points, it is called an ideal triangle. All ideal triangles are the same size, and their inside angles are zero. They also have an infinite perimeter.

If all the corners of a quadrilateral are ideal points, it is called an ideal quadrilateral. Not all ideal quadrilaterals are the same, but they all share some properties. Like ideal triangles, their inside angles are zero, and they also have an infinite perimeter. An ideal square is a special type of ideal quadrilateral where the two diagonals cross at right angles. Ideal shapes can be split into ideal triangles to find their area.

Representations in models of hyperbolic geometry

In the Klein disk model and the Poincaré disk model of the hyperbolic plane, ideal points lie on the unit circle or unit sphere, which forms the boundary that cannot be reached. When we project the same hyperbolic line onto both the Klein disk model and the Poincaré disk model, the line passes through the same two ideal points in both models.

In the Poincaré half-plane model, ideal points are located on the boundary axis, with an additional ideal point approached by rays parallel to the positive y-axis. However, in the hyperboloid model, there are no ideal points at all.