In mathematics, the upper half-plane, ⁠⁠ is the set of points ⁠⁠ in the Cartesian plane with ⁠⁠ The lower half-plane is the set of points ⁠⁠ with ⁠⁠ instead. Each is an example of two-dimensional half-space.

The affine transformations of the upper half-plane include

1. shifts , , and
2. 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 ⁠⁠ to the center of ⁠⁠

Definition: .

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

Proposition: ⁠⁠ ⁠⁠ in ⁠⁠ and ⁠⁠ are collinear points.

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

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.

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.

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.

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

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