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A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. The open (closed) upper half-space is the half-space of all (x 1, x 2, ..., x n) such that x n > 0
The lower half-plane is the set of points (,) with < instead. Arbitrary oriented half-planes can be obtained via a planar rotation . Half-planes are an example of two-dimensional half-space .
In complex analysis, the (open) right half-plane is the set of all points in the complex plane whose real part is strictly positive, that is, ...
Half-space (geometry), either of the two parts into which a plane divides Euclidean space (Poincaré) Half-space model, a model of hyperbolic geometry using a Euclidean half-space; Siegel upper half-space, a set of complex matrices with positive definite imaginary part; Half-space (punctuation), a spacing character half the width of a regular space
A convex set (in pink), a supporting hyperplane of (the dashed line), and the supporting half-space delimited by the hyperplane which contains (in light blue). In geometry , a supporting hyperplane of a set S {\displaystyle S} in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is a hyperplane that has both of the following two properties ...
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, , that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition.