Half-plane

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In mathematics, the upper half-plane, ⁠ H , {\displaystyle {\mathcal {H}},} {\displaystyle {\mathcal {H}},}⁠ is the set of points ⁠ ( x , y ) {\displaystyle (x,y)} {\displaystyle (x,y)}⁠ in the Cartesian plane with ⁠ y > 0. {\displaystyle y>0.} {\displaystyle y>0.}⁠ The lower half-plane is the set of points ⁠ ( x , y ) {\displaystyle (x,y)} {\displaystyle (x,y)}⁠ with ⁠ y < 0 {\displaystyle y<0} {\displaystyle y<0}⁠ instead. Arbitrarily 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.

Affine geometry

The affine transformations of the upper half-plane include

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

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

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

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

Inversive geometry

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

⁠ Z {\displaystyle {\mathcal {Z}}} {\displaystyle {\mathcal {Z}}}⁠ can be recognized as the circle of radius ⁠ 1 2 {\displaystyle {\tfrac {1}{2}}} {\displaystyle {\tfrac {1}{2}}}⁠ centered at ⁠ ( 1 2 , 0 ) , {\displaystyle {\bigl (}{\tfrac {1}{2}},0{\bigr )},} {\displaystyle {\bigl (}{\tfrac {1}{2}},0{\bigr )},}⁠ and as the polar plot of ρ ( θ ) = cos ⁡ θ . {\displaystyle \rho (\theta )=\cos \theta .} {\displaystyle \rho (\theta )=\cos \theta .}

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

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

Metric geometry

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

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:

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

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

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

Generalizations

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

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

See also

References