Definition:Half-Plane/Closed

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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 $y \le 0$:

$\HH_{\text {CL} } := \set {\tuple {x, y}: y \in \R_{\le 0} }$


Also see