The lower half-plane is the set of points
Arbitrary 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.
The affine transformations of the upper half-plane include Proposition: Let
be semicircles in the upper half-plane with centers on the boundary.
can be recognized as the circle of radius
in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from
either intersects the boundary or is parallel to it.
lie on a ray perpendicular to the boundary and logarithmic measure can be used to define a distance that is invariant under dilation.
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
In consequence, the upper half-plane becomes a metric space.
The generic name of this metric space is the hyperbolic 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: The term arises from a common visualization of the complex number
in the plane endowed with Cartesian coordinates.
axis is oriented vertically, the "upper half-plane" corresponds to the region above the
It is the domain of many functions of interest in complex analysis, especially modular forms.
The lower half-plane, defined by
(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
It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions.
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.
One natural generalization in differential geometry is hyperbolic
the maximally symmetric, simply connected,
-dimensional Riemannian manifold with constant sectional curvature
In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product
Yet another space interesting to number theorists is the Siegel upper half-space
which is the domain of Siegel modular forms.