Definition:Half-Plane/Closed
< Definition:Half-Plane(Redirected from Definition:Closed Half-Plane)
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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 closed half-plane if and only if $\HH$ includes $\LL$.
Instances
Closed Left Half-Plane
The closed left half-plane $\HH_{\text {CL} }$ is the area of $\PP$ on the left of and including $\LL$.
That is, where $x \le 0$:
- $\HH_{\text {CL} } := \set {\tuple {x, y}: x \in \R_{\le 0} }$
Closed Right Half-Plane
The closed right half-plane $\HH_{\text {CR} }$ is the area of $\PP$ on the right of and including $\LL$.
That is, where $x \ge 0$:
- $\HH_{\text {CR} } := \set {\tuple {x, y}: x \in \R_{\ge 0} }$
Closed Upper Half-Plane
The closed upper half-plane $\HH_{\text {CU} }$ is the area of $\PP$ above and including $\LL$.
That is, where $y \ge 0$:
- $\HH_{\text {CU} } := \set {\tuple {x, y}: y \in \R_{\ge 0} }$
Closed Lower Half-Plane
The closed lower half-plane $\HH_{\text {CL} }$ is the area of $\PP$ below and including $\LL$.
That is, where $y \le 0$:
- $\HH_{\text {CL} } := \set {\tuple {x, y}: y \in \R_{\le 0} }$