# Upper half-plane

(Redirected from Half-plane)

In mathematics, the upper half-plane, $\,{\mathcal {H}}\,$ , is the set of points $(x,y)$ in the Cartesian plane with $y>0$ . The lower half-plane is defined similarly, by requiring that $y$ be negative instead. Each is an example of two-dimensional half-space.

## Affine geometry

The affine transformations of the upper half-plane include

1. shifts $(x,y)\mapsto (x+c,y)$ , $c\in \mathbb {R}$ , and
2. dilations $(x,y)\mapsto (\lambda x,\lambda y)$ , $\lambda >0$ .

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

Proof: First shift the center of $A$ to $(0,0)$ . Then take $\lambda =({\text{diameter of}}\ B)/({\text{diameter of}}\ A)$ and dilate. Then shift $(0,0)$ the center of $B$ .

Definition: ${\mathcal {Z}}:=\left\{\left(\cos ^{2}(\theta ),{\tfrac {1}{2}}\sin(2\theta )\right)\mid 0<\theta <\pi \right\}$ .

${\mathcal {Z}}$ can be recognized as the circle of radius $1/2$ centered at $(1/2,0)$ , and as the polar plot of $\rho (\theta )=\cos(\theta )$ .

Proposition: $(0,0)$ , $\rho (\theta )\in {\mathcal {Z}}$ , and $(1,\tan(\theta ))$ are collinear points.

In fact, ${\mathcal {Z}}$ is the reflection of the line ${\bigl \{}(1,y)\mid y>0{\bigr \}}$ in the unit circle. Indeed, the diagonal from $(0,0)$ to $(1,\tan(\theta ))$ has squared length $1+\tan ^{2}(\theta )=\sec ^{2}(\theta )$ , so that $\rho (\theta )=\cos(\theta )$ is the reciprocal of that length.

## Metric geometry

The distance between any two points $p$ and $q$ in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from $p$ to $q$ either intersects the boundary or is parallel to it. In the latter case $p$ and $q$ 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 $p$ and $q$ 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 ${\mathcal {Z}}$ . Distances on ${\mathcal {Z}}$ can be defined using the correspondence with points on ${\bigl \{}(1,y)\mid y>0{\bigr \}}$ 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

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:

${\mathcal {H}}:=\{x+iy\mid y>0;\ x,y\in \mathbb {R} \}.$ The term arises from a common visualization of the complex number $x+iy$ as the point $(x,y)$ in the plane endowed with Cartesian coordinates. When the $y$ axis is oriented vertically, the "upper half-plane" corresponds to the region above the $x$ axis and thus complex numbers for which $y>0$ .

It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by $y<0$ is equally good, but less used by convention. The open unit disk ${\mathcal {D}}$ (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to ${\mathcal {H}}$ (see "Poincaré metric"), meaning that it is usually possible to pass between ${\mathcal {H}}$ and ${\mathcal {D}}$ .

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

One natural generalization in differential geometry is hyperbolic $n$ -space ${\mathcal {H}}^{n}$ , the maximally symmetric, simply connected, $n$ -dimensional Riemannian manifold with constant sectional curvature $-1$ . In this terminology, the upper half-plane is ${\mathcal {H}}^{2}$ since it has real dimension $2$ .

In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product ${\mathcal {H}}^{n}$ of $n$ copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space ${\mathcal {H}}_{n}$ , which is the domain of Siegel modular forms.