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Concept
Upper half-plane
Summary
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.
The affine transformations of the upper half-plane include
shifts , , and
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.
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.
Official source
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