Definition:Half-Plane

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Definition

Let $P$ denote the plane.

Let $L$ denote an infinite straight line in $P$.


Then each of the areas of $P$ on either side of $L$ is a half-plane.


Edge of Half-Plane

Let $H$ denote one of the half-planes into which $L$ divides $P$.

Then $L$ is called the edge of $H$.


Open or Closed

Let $H$ be a half-plane whose edge is $L$.


Open Half-Plane

$H$ is an open half-plane if and only if $H$ does not include $L$.


Closed Half-Plane

$H$ is an closed half-plane if and only if $H$ includes $L$.


Left Half-Plane

Open Left Half-Plane

The open left half-plane $H_{\text {OL} }$ is the area of $P$ on the left of $L$.

That is, where $x < 0$:

$H_{\text {OL} } := \set {\tuple {x, y}: x \in \R_{<0} }$


Closed Left Half-Plane

The closed left half-plane $H_{\text {CL} }$ is the area of $P$ on the left of and including $L$.

That is, where $x \le 0$:

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


Right Half-Plane

Open Right Half-Plane

The open right half-plane $H_{\text {OR} }$ is the area of $P$ on the right of $L$.

That is, where $x > 0$:

$H_{\text {OR} } := \set {\tuple {x, y}: x \in \R_{>0} }$


Closed Right Half-Plane

The closed right half-plane $H_{\text {CR} }$ is the area of $P$ on the right of and including $L$.

That is, where $x \ge 0$:

$H_{\text {CR} } := \set {\tuple {x, y}: x \in \R_{\ge 0} }$


Upper Half-Plane

Open Upper Half-Plane

The open upper half-plane $H_{\text {OU} }$ is the area of $P$ above $L$.

That is, where $y > 0$:

$H_{\text {OU} } := \set {\tuple {x, y}: y \in \R_{> 0} }$


Closed Upper Half-Plane

The closed upper half-plane $H_{\text {CU} }$ is the area of $P$ above and including $L$.

That is, where $y \ge 0$:

$H_{\text {CU} } := \set {\tuple {x, y}: y \in \R_{\ge 0} }$


Lower Half-Plane

Open Lower Half-Plane

The open lower half-plane $H_{\text {OL} }$ is the area of $P$ below $L$.

That is, where $y < 0$:

$H_{\text {OL} } := \set {\tuple {x, y}: y \in \R_{< 0} }$


Closed Lower Half-Plane

The closed lower half-plane $H_{\text {CL} }$ is the area of $P$ below and including $L$.

That is, where $y \le 0$:

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


Sources