Upper half-plane

In mathematics, the upper half-plane, ⁠ $ℋ,$ ⁠ is the set of points ⁠ $(x, y)$ ⁠ in the Cartesian plane with ⁠ $y > 0.$ ⁠ The lower half-plane is the set of points ⁠ $(x, y)$ ⁠ with ⁠ $y < 0$ ⁠ instead. Arbitrarily oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants.

Affine geometry

The affine transformations of the upper half-plane include

shifts $(x, y) \mapsto (x + c, y)$ , $c \in ℝ$ , and
dilations $(x, y) \mapsto (λ x, λ y)$ , $λ > 0.$

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

λ = (diameter of B) / (diameter of A)

and dilate. Then shift ⁠ $(0, 0)$ ⁠ to the center of ⁠ $B .$ ⁠

Inversive geometry

Definition: $𝒵 := {(\cos^{2} θ, \frac{1}{2} \sin 2 θ) ∣ 0 < θ < π}$ .

⁠ $𝒵$ ⁠ can be recognized as the circle of radius ⁠ $\frac{1}{2}$ ⁠ centered at ⁠ $(\frac{1}{2}, 0),$ ⁠ and as the polar plot of $ρ (θ) = \cos θ .$

Proposition: ⁠ $(0, 0),$ ⁠ ⁠ $ρ (θ)$ ⁠ in ⁠ $𝒵,$ ⁠ and ⁠ $(1, \tan θ)$ ⁠ are collinear points.

In fact, $𝒵$ is the inversion of the line ${(1, y) ∣ y > 0}$ in the unit circle. Indeed, the diagonal from ⁠ $(0, 0)$ ⁠ to ⁠ $(1, \tan θ)$ ⁠ has squared length $1 + \tan^{2} θ = \sec^{2} θ$ , so that $ρ (θ) = \cos θ$ 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 ⁠ $𝒵 .$ ⁠ Distances on ⁠ $𝒵$ ⁠ can be defined using the correspondence with points on ${(1, y) ∣ y > 0}$ 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:

ℋ := {x + i y ∣ y > 0; x, y \in ℝ} .

The term arises from a common visualization of the complex number $x + i y$ 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 ⁠ $𝒟$ ⁠ (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to ⁠ $ℋ$ ⁠ (see "Poincaré metric"), meaning that it is usually possible to pass between ⁠ $ℋ$ ⁠ and ⁠ $𝒟 .$ ⁠

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 ⁠ $ℋ^{n},$ ⁠ the maximally symmetric, simply connected, ⁠ $n$ ⁠-dimensional Riemannian manifold with constant sectional curvature $- 1$ . In this terminology, the upper half-plane is ⁠ $ℋ^{2}$ ⁠ 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 ⁠ $ℋ^{n}$ ⁠ of ⁠ $n$ ⁠ copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space ⁠ $ℋ_{n},$ ⁠ which is the domain of Siegel modular forms.