Hyperbolic plane isometry

Add caption to hyperbolic isometry picture.

cjerdonek authored on Dec. 2, 2020, 4:13 a.m.

Review changes View this version of the page

You can also use the and keys

~~Deletions are marked like this~~	Additions are marked like this
An isometry of the hyperbolic plane is a mapping of the hyperbolic plane to itself that preserves the underlying hyperbolic geometry (e.g. distances and angles). An isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. We first focus on orientation-preserving isometries. There are three types of orientation-preserving isometries of the hyperbolic plane: hyperbolic, elliptic, and parabolic. (This terminology can be confusing at times because "hyperbolic isometry" can be used to mean both any isometry of the hyperbolic plane, as well as only those isometries that have hyperbolic type. In general, a reader will have to pay attention to the context to know which meaning is being used.) One way to tell the type of an isometry is by its fixed points. We will discuss each type below. Hyperbolic isometry A hyperbolic isometry has the property that it has two fixed points, and those points are at the boundary at infinity. A hyperbolic isometry is sometimes called a "translation." Elliptic isometry An elliptic isometry has the property that it has one fixed point in the interior of the hyperbolic plane. An elliptic isometry is sometimes called a "rotation." Parabolic isometry A parabolic isometry has the property that it has one fixed point on the boundary of the hyperbolic plane.	An isometry of the hyperbolic plane is a mapping of the hyperbolic plane to itself that preserves the underlying hyperbolic geometry (e.g. distances and angles). An isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. We first focus on orientation-preserving isometries. There are three types of orientation-preserving isometries of the hyperbolic plane: hyperbolic, elliptic, and parabolic. (The terminology can be confusing at times because "hyperbolic isometry" can be used to mean both any isometry of the hyperbolic plane, as well as only those isometries that have hyperbolic type. In general, a reader will have to pay attention to the context to know which meaning is being used.) The main way to tell the type of an isometry is by its fixed points. We will discuss each type below. Hyperbolic isometry A hyperbolic isometry has the property that it has exactly two fixed points. These two points are always at the boundary at infinity. A hyperbolic isometry is sometimes called a "translation." A hyperbolic isometry is determined by (1) its two fixed points, or equivalently the geodesic it fixes, (2) a direction, or orientation of that geodesic, and (3) its translation length. This figure shows a family of hyperbolic isometries animated with increasing translation length. Both sides show the same isometries, with the left using the Klein model, and the right using the Poincaré disk model. Along the geodesics, the dots are spaced distance 0.25 apart in the Klein model, and 0.5 apart in the Poincare disk model. Elliptic isometry An elliptic isometry has the property that it has one fixed point in the interior of the hyperbolic plane. An elliptic isometry is sometimes called a "rotation." Parabolic isometry A parabolic isometry has the property that it has one fixed point on the boundary of the hyperbolic plane.

~~Deletions are marked like this~~

Additions are marked like this

An isometry of the hyperbolic plane is a mapping of the hyperbolic plane to itself that preserves the underlying hyperbolic geometry (e.g. distances and angles).

An isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. We first focus on orientation-preserving isometries.

There are three types of orientation-preserving isometries of the hyperbolic plane: hyperbolic, elliptic, and parabolic. (This terminology can be confusing at times because "hyperbolic isometry" can be used to mean both any isometry of the hyperbolic plane, as well as only those isometries that have hyperbolic type. In general, a reader will have to pay attention to the context to know which meaning is being used.)

One way to tell the type of an isometry is by its fixed points. We will discuss each type below.

Hyperbolic isometry

A hyperbolic isometry has the property that it has two fixed points, and those points are at the boundary at infinity. A hyperbolic isometry is sometimes called a "translation."

Elliptic isometry

An elliptic isometry has the property that it has one fixed point in the interior of the hyperbolic plane. An elliptic isometry is sometimes called a "rotation."

Parabolic isometry

A parabolic isometry has the property that it has one fixed point on the boundary of the hyperbolic plane.

An isometry of the hyperbolic plane is a mapping of the hyperbolic plane to itself that preserves the underlying hyperbolic geometry (e.g. distances and angles).

An isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. We first focus on orientation-preserving isometries.

There are three types of orientation-preserving isometries of the hyperbolic plane: hyperbolic, elliptic, and parabolic. (The terminology can be confusing at times because "hyperbolic isometry" can be used to mean both any isometry of the hyperbolic plane, as well as only those isometries that have hyperbolic type. In general, a reader will have to pay attention to the context to know which meaning is being used.)

The main way to tell the type of an isometry is by its fixed points. We will discuss each type below.

Hyperbolic isometry

A hyperbolic isometry has the property that it has exactly two fixed points. These two points are always at the boundary at infinity. A hyperbolic isometry is sometimes called a "translation." A hyperbolic isometry is determined by (1) its two fixed points, or equivalently the geodesic it fixes, (2) a direction, or orientation of that geodesic, and (3) its translation length.

This figure shows a family of hyperbolic isometries animated with increasing translation length. Both sides show the same isometries, with the left using the Klein model, and the right using the Poincaré disk model. Along the geodesics, the dots are spaced distance 0.25 apart in the Klein model, and 0.5 apart in the Poincare disk model.

Elliptic isometry

An elliptic isometry has the property that it has one fixed point in the interior of the hyperbolic plane. An elliptic isometry is sometimes called a "rotation."

Parabolic isometry

A parabolic isometry has the property that it has one fixed point on the boundary of the hyperbolic plane.

cjerdonek / Hyperbolic plane isometry e77737c8f73b4ccb559b771d494a14a3e1de0a2f

Hyperbolic isometry

Elliptic isometry

Parabolic isometry

Hyperbolic isometry

Elliptic isometry

Parabolic isometry