Composition of projections remains the same

Question

Consider a plane $π : Ax + By + Cz + D = 0$ and a line $d : \frac{x − x_0} p = \frac{y − y_0} q = \frac{z − z_0} r$ . If $π \nparallel d$, show that:$p_{\pi,d}\circ p_{\pi,d} = p_{\pi,d}$. I have found the next formula but I don't know how to use is.

Answer 1

Instead of trying to use an unwieldy formula that you don’t understand, work with the definition of the projection: the projection of a point $\mathbf x$ is the intersection of the line through $\mathbf x$ parallel to $d$ with the plane $\pi$. It should be obvious the the projection of any point on $\mathbf\pi$ is the point itself, therefore $p_{\mathbf\pi,d}(p_{\mathbf\pi,d}(\mathbf x))=p_{\mathbf\pi,d}(\mathbf x)$.

If you must work with that formula that you looked up somewhere, I’d suggest packaging it up into the $4\times4$ matrix $$P = \frac1{Ap+Bq+Cr}\begin{bmatrix}Bq+Cr&-Bp&-Cp&-Dp\\-Aq&Ap+Cr&-Cq&-Dq\\-Ar&-Br&Ap+Bq&-Dr\\0&0&0&1\end{bmatrix}$$ and then verifying that $P^2=P$.