# Upper half-plane

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In mathematics, the **upper half-plane**, , is the set of points in the Cartesian plane with
. The **lower half-plane** is defined similarly, by requiring that be negative instead. Each is an example of two-dimensional half-space.

## Affine geometry[edit]

The affine transformations of the upper half-plane include

- shifts , , and
- dilations , .

**Proposition:** Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes
to .

- Proof: First shift the center of to . Then take

and dilate. Then shift the center of .

**Definition:** .

can be recognized as the circle of radius centered at , and as the polar plot of .

**Proposition:** , , and are collinear points.

In fact, is the reflection of the line in the unit circle. Indeed, the diagonal from to has squared length , so that is the reciprocal of that length.

## Metric geometry[edit]

The distance between any two points and in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from to either intersects the boundary or is parallel to it. In the latter case and lie on a ray perpendicular to the boundary and logarithmic measure can be used to define a distance that is invariant under dilation. In the former case and lie on a circle centered at the intersection of their perpendicular bisector and the boundary. By the above proposition this circle can be moved by affine motion to . Distances on can be defined using the correspondence with points on and logarithmic measure on this ray. In consequence, the upper half-plane becomes a metric space. The generic name of this metric space is the hyperbolic plane. In terms of the models of hyperbolic geometry, this model is frequently designated the Poincaré half-plane model.

## Complex plane[edit]

Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part:

The term arises from a common visualization of the complex number as the point in the plane endowed with Cartesian coordinates. When the axis is oriented vertically, the "upper half-plane" corresponds to the region above the axis and thus complex numbers for which .

It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by is equally good, but less used by convention. The open unit disk (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to (see "Poincaré metric"), meaning that it is usually possible to pass between and .

It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric on the space.

The uniformization theorem for surfaces states that the **upper half-plane** is the universal covering space of surfaces with constant negative Gaussian curvature.

The **closed upper half-plane** is the union of the upper half-plane and the real axis. It is the closure of the upper half-plane.

## Generalizations[edit]

One natural generalization in differential geometry is hyperbolic -space , the maximally symmetric, simply connected, -dimensional Riemannian manifold with constant sectional curvature . In this terminology, the upper half-plane is since it has real dimension .

In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product of copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space , which is the domain of Siegel modular forms.

## See also[edit]

- Cusp neighborhood
- Extended complex upper-half plane
- Fuchsian group
- Fundamental domain
- Half-space
- Kleinian group
- Modular group
- Riemann surface
- Schwarz–Ahlfors–Pick theorem
- Moduli stack of elliptic curves