Definition:Half-Plane

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Definition

Let $\PP$ denote the plane.

Let $\LL$ denote an infinite straight line in $\PP$.


Then each of the areas of $\PP$ on either side of $\LL$ is a half-plane.


Edge of Half-Plane

Let $\HH$ denote one of the half-planes into which $\LL$ divides $\PP$.

Then $\LL$ is called the edge of $\HH$.


Sign of Half-Plane

Let $\LL$ be a straight line embedded in a cartesian plane $\CC$, given by the equation:

$l x + m y + n = 0$

Let $\HH_1$ and $\HH_2$ be the half-planes into which $\LL$ divides $\CC$.

Let $u: \CC \to \R$ be the real-valued function on the points in $\CC$ defined as follows:

$\forall P = \tuple {x_1, y_1} \in \CC: \map u P = l x_1 + m y_1 + n$


The sign of the half-plane $\HH_j$ is the sign of the value of $\map u Q$ for all $Q \in \HH_j$, for $j \in \set {1, 2}$.


Open or Closed

Let $\HH$ be a half-plane whose edge is $\LL$.


Open Half-Plane

$\HH$ is an open half-plane if and only if $\HH$ does not include $\LL$.


Closed Half-Plane

$\HH$ is an closed half-plane if and only if $\HH$ includes $\LL$.


Left Half-Plane

Let $\LL$ be the $y$-axis.


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 $x < 0$:

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


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 $x \le 0$:

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


Right Half-Plane

Let $\LL$ be the $y$-axis.


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 $x > 0$:

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


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 $x \ge 0$:

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


Upper Half-Plane

Let $\LL$ be the $x$-axis.


Open Upper Half-Plane

The open upper half-plane $\HH_{\text {OU} }$ is the area of $\PP$ above $\LL$.

That is, where $y > 0$:

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


Closed Upper Half-Plane

The closed upper half-plane $\HH_{\text {CU} }$ is the area of $\PP$ above and including $\LL$.

That is, where $y \ge 0$:

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


Lower Half-Plane

Let $\LL$ be the $x$-axis.


Open Lower Half-Plane

The open lower half-plane $\HH_{\text {OL} }$ is the area of $\PP$ below $\LL$.

That is, where $y < 0$:

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


Closed Lower Half-Plane

The closed lower half-plane $\HH_{\text {CL} }$ is the area of $\PP$ below and including $\LL$.

That is, where $y \le 0$:

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


Sources

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