Hyperbolic plane isometry

cjerdonek authored on Dec. 2, 2020, 6:19 p.m.

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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). The isometries of the hyperbolic plane form a group under composition. An isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. We first focus on orientation-preserving isometries. Orientation-preserving isometries The orientation-preserving isometries of the hyperbolic plane form a subgroup. This subgroup is equal to \(PSL(2, \mathbf{RP})\). There are three types of orientation-preserving isometries of the hyperbolic plane: hyperbolic, elliptic, and parabolic. (This terminology can be confusing at first because "hyperbolic isometry" can be used both to mean any isometry of the hyperbolic plane, as well as only those isometries that have hyperbolic type. In general, readers will have to pay attention to the context to know which meaning is being used.) There is also a fourth type of isometry that is often not mentioned because it is trivial: the identity isometry. Alternatively, the identity isometry can be viewed as a degenerate form of the other three types. Whether an isometry is hyperbolic, elliptic, parabolic, or the identity is invariant under conjugacy in the group. The main way to tell the type of an isometry is by its fixed points. For example, the identity isometry is the isometry that fixes all points. We will discuss the remaining types below. Hyperbolic isometry A hyperbolic isometry is distinguished by the fact that it has exactly two fixed points. These two points are always at the boundary at infinity. A hyperbolic isometry fixes (as a set) the geodesic connecting those two points. 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. Orientation-reversing isometries TODO	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). The isometries of the hyperbolic plane form a group under composition. An isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. We first focus on orientation-preserving isometries. Orientation-preserving isometries The orientation-preserving isometries of the hyperbolic plane form a subgroup. This subgroup is equal to \(PSL(2, \mathbf{R})\). There are three types of orientation-preserving isometries of the hyperbolic plane: hyperbolic, elliptic, and parabolic. (This terminology can be confusing at first because "hyperbolic isometry" can be used both to mean any isometry of the hyperbolic plane, as well as only those isometries that have hyperbolic type. In general, readers will have to pay attention to the context to know which meaning is being used.) There is also a fourth type of isometry that is often not mentioned because it is trivial: the identity isometry. Alternatively, the identity isometry can be viewed as a degenerate form of the other three types. Whether an isometry is hyperbolic, elliptic, parabolic, or the identity is invariant under conjugacy in the group. The main way to tell the type of an isometry is by its fixed points. For example, the identity isometry is the isometry that fixes all points. We will discuss the remaining types below. Hyperbolic isometry A hyperbolic isometry is distinguished by the fact that it has exactly two fixed points. These two points are always at the boundary at infinity. A hyperbolic isometry fixes (as a set) the geodesic connecting those two points. 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. Orientation-reversing isometries TODO

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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).

The isometries of the hyperbolic plane form a group under composition.

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

Orientation-preserving isometries

The orientation-preserving isometries of the hyperbolic plane form a subgroup. This subgroup is equal to \(PSL(2, \mathbf{RP})\).

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

There is also a fourth type of isometry that is often not mentioned because it is trivial: the identity isometry. Alternatively, the identity isometry can be viewed as a degenerate form of the other three types.

Whether an isometry is hyperbolic, elliptic, parabolic, or the identity is invariant under conjugacy in the group.

The main way to tell the type of an isometry is by its fixed points. For example, the identity isometry is the isometry that fixes all points. We will discuss the remaining types below.

Hyperbolic isometry

A hyperbolic isometry is distinguished by the fact that it has exactly two fixed points. These two points are always at the boundary at infinity. A hyperbolic isometry fixes (as a set) the geodesic connecting those two points.

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.

Orientation-reversing isometries

TODO

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).

The isometries of the hyperbolic plane form a group under composition.

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

Orientation-preserving isometries

The orientation-preserving isometries of the hyperbolic plane form a subgroup. This subgroup is equal to \(PSL(2, \mathbf{R})\).

Whether an isometry is hyperbolic, elliptic, parabolic, or the identity is invariant under conjugacy in the group.

The main way to tell the type of an isometry is by its fixed points. For example, the identity isometry is the isometry that fixes all points. We will discuss the remaining types below.

Hyperbolic isometry

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.

Orientation-reversing isometries

TODO

cjerdonek / Hyperbolic plane isometry be4652c3a595e5ba499beb47cc38295ca9b0617c

Orientation-preserving isometries

Hyperbolic isometry

Elliptic isometry

Parabolic isometry

Orientation-reversing isometries

Orientation-preserving isometries

Hyperbolic isometry

Elliptic isometry

Parabolic isometry

Orientation-reversing isometries