Upper half-plane


In mathematics, the upper half-plane, , is the set of points in the Cartesian plane with . The lower half-plane is defined similarly, by requiring that be negative instead. Each is an example of two-dimensional half-space.

Affine geometry edit

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   the center of  .

Definition:  .

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

Proposition:  ,  , and   are collinear points.

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

Metric geometry edit

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.

Complex plane edit

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.

Generalizations edit

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.

See also edit

References edit

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