Definition:Perpendicular Projection

Definition

Let $\PP$ denote the plane.

Let $L$ denote a straight line in $\PP$.

For all $p \in \PP$, let $K_p$ denote the straight line through $P$ perpendicular to $L$.

Let $p_L$ denote the point on $L$ where $K_p$ intersects $L$.

Let $\pi_L: \PP \to L$ denote the mapping defined as:

$\forall p \in \PP: \map {\pi_L} p = p_L$

That is, $\pi_L$ sends every point $p$ in $\PP$ to the foot of the perpendicular from $p$ to $L$.

$\pi_L$ is called the perpendicular projection of the plane onto $L$.