The affine transformations of the upper half-plane include

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

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

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

and dilate. Then shift ${\displaystyle (0,0)}$  the center of ${\displaystyle B}$ .

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

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

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

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

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

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:

${\displaystyle {\mathcal {H}}:=\{x+iy\mid y>0;\ x,y\in \mathbb {R} \}.}$ 

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

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

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.

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

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

• 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

• Weisstein, Eric W. "Upper Half-Plane". MathWorld.