Ideal point

An ideal point is a special idea used in a type of math called hyperbolic geometry. This geometry is different from the flat geometry we see around us. In hyperbolic geometry, ideal points exist outside the main space we study. They help us understand how lines behave when they go very 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 is outside the main space we study. The distance from any ideal point to other points is always infinite. These points help create shapes called horocycles and horoballs. When two horocycles share the same ideal point, 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. Their inside angles are zero, and they 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 share some properties. Like ideal triangles, their inside angles are zero, and they 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 are found on the unit circle or unit sphere. These points mark the edge that we cannot reach. When we draw the same hyperbolic line in both the Klein disk model and the Poincaré disk model, the line goes through the same two ideal points in each model.

In the Poincaré half-plane model, ideal points sit on the boundary axis. There is also one more ideal point that we get when we follow lines that go parallel to the positive y-axis. But in the hyperboloid model, there are no ideal points at all.