Definition:Half-Plane/Sign

Definition
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 $j \in \set {1, 2}$.

Also see

 * Sign of Half-Plane is Well-Defined