Definition:Half-Plane/Open
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Definition
Let $\PP$ denote the plane.
Let $\LL$ denote an infinite straight line in $\PP$.
Let $\HH$ be a half-plane whose edge is $\LL$.
$\HH$ is an open half-plane if and only if $\HH$ does not include $\LL$.
Instances
Open Left Half-Plane
The open left half-plane $\HH_{\text {OL} }$ is the area of $\PP$ on the left of $\LL$.
That is, where $x < 0$:
- $\HH_{\text {OL} } := \set {\tuple {x, y}: x \in \R_{<0} }$
Open Right Half-Plane
The open right half-plane $\HH_{\text {OR} }$ is the area of $\PP$ on the right of $\LL$.
That is, where $x > 0$:
- $\HH_{\text {OR} } := \set {\tuple {x, y}: x \in \R_{>0} }$
Open Upper Half-Plane
The open upper half-plane $\HH_{\text {OU} }$ is the area of $\PP$ above $\LL$.
That is, where $y > 0$:
- $\HH_{\text {OU} } := \set {\tuple {x, y}: y \in \R_{> 0} }$
Open Lower Half-Plane
The open lower half-plane $\HH_{\text {OL} }$ is the area of $\PP$ below $\LL$.
That is, where $y < 0$:
- $\HH_{\text {OL} } := \set {\tuple {x, y}: y \in \R_{< 0} }$