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

The affine transformations of the upper half-plane include

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

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

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

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

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

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