Upper half-plane
In mathematics, the upper half-plane, [math]\displaystyle{ \,\mathcal{H}\, }[/math], is the set of points [math]\displaystyle{ (x,y) }[/math] in the Cartesian plane with [math]\displaystyle{ y \gt 0 }[/math]. The lower half-plane is defined similarly, by requiring that [math]\displaystyle{ y }[/math] be negative instead. Each is an example of two-dimensional half-space.
Affine geometry
The affine transformations of the upper half-plane include
- shifts [math]\displaystyle{ (x,y)\mapsto (x+c,y) }[/math], [math]\displaystyle{ c\in\mathbb{R} }[/math], and
- dilations [math]\displaystyle{ (x,y)\mapsto (\lambda x,\lambda y) }[/math], [math]\displaystyle{ \lambda \gt 0 }[/math].
Proposition: Let [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes [math]\displaystyle{ A }[/math] to [math]\displaystyle{ B }[/math].
- Proof: First shift the center of [math]\displaystyle{ A }[/math] to [math]\displaystyle{ (0,0) }[/math]. Then take [math]\displaystyle{ \lambda=(\text{diameter of}\ B)/(\text{diameter of}\ A) }[/math]
and dilate. Then shift [math]\displaystyle{ (0,0) }[/math] the center of [math]\displaystyle{ B }[/math].
Definition: [math]\displaystyle{ \mathcal{Z} := \left\{\left( \cos^2(\theta),\tfrac{1}{2} \sin(2\theta) \right) \mid 0 \lt \theta \lt \pi \right\} }[/math].
[math]\displaystyle{ \mathcal{Z} }[/math] can be recognized as the circle of radius [math]\displaystyle{ 1/2 }[/math] centered at [math]\displaystyle{ (1/2,0) }[/math], and as the polar plot of [math]\displaystyle{ \rho(\theta) = \cos(\theta) }[/math].
Proposition: [math]\displaystyle{ (0,0) }[/math], [math]\displaystyle{ \rho(\theta) \in \mathcal{Z} }[/math], and [math]\displaystyle{ (1,\tan(\theta)) }[/math] are collinear points.
In fact, [math]\displaystyle{ \mathcal{Z} }[/math] is the reflection of the line [math]\displaystyle{ \bigl\{(1, y) \mid y \gt 0 \bigr\} }[/math] in the unit circle. Indeed, the diagonal from [math]\displaystyle{ (0,0) }[/math] to [math]\displaystyle{ (1, \tan(\theta)) }[/math] has squared length [math]\displaystyle{ 1 + \tan^2(\theta) = \sec^2(\theta) }[/math], so that [math]\displaystyle{ \rho(\theta) = \cos(\theta) }[/math] is the reciprocal of that length.
Metric geometry
The distance between any two points [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from [math]\displaystyle{ p }[/math] to [math]\displaystyle{ q }[/math] either intersects the boundary or is parallel to it. In the latter case [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] 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 [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] 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 [math]\displaystyle{ \mathcal{Z} }[/math]. Distances on [math]\displaystyle{ \mathcal{Z} }[/math] can be defined using the correspondence with points on [math]\displaystyle{ \bigl\{(1, y)\mid y \gt 0 \bigr\} }[/math] 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:
- [math]\displaystyle{ \mathcal{H} := \{x + iy \mid y \gt 0;\ x, y \in \mathbb{R} \} . }[/math]
The term arises from a common visualization of the complex number [math]\displaystyle{ x+iy }[/math] as the point [math]\displaystyle{ (x,y) }[/math] in the plane endowed with Cartesian coordinates. When the [math]\displaystyle{ y }[/math] axis is oriented vertically, the "upper half-plane" corresponds to the region above the [math]\displaystyle{ x }[/math] axis and thus complex numbers for which [math]\displaystyle{ y \gt 0 }[/math].
It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by [math]\displaystyle{ y \lt 0 }[/math] is equally good, but less used by convention. The open unit disk [math]\displaystyle{ \mathcal{D} }[/math] (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to [math]\displaystyle{ \mathcal{H} }[/math] (see "Poincaré metric"), meaning that it is usually possible to pass between [math]\displaystyle{ \mathcal{H} }[/math] and [math]\displaystyle{ \mathcal{D} }[/math].
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 [math]\displaystyle{ n }[/math]-space [math]\displaystyle{ \mathcal{H}^n }[/math], the maximally symmetric, simply connected, [math]\displaystyle{ n }[/math]-dimensional Riemannian manifold with constant sectional curvature [math]\displaystyle{ -1 }[/math]. In this terminology, the upper half-plane is [math]\displaystyle{ \mathcal{H}^2 }[/math] since it has real dimension [math]\displaystyle{ 2 }[/math].
In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product [math]\displaystyle{ \mathcal{H}^n }[/math] of [math]\displaystyle{ n }[/math] copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space [math]\displaystyle{ \mathcal{H}_n }[/math], which is the domain of Siegel modular forms.
See also
- 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
References
de:Obere Halbebene it:Semipiano
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