Fixed Points of Projection in Plane

Theorem
Let $M$ and $N$ be distinct lines in the plane.


 * Projection-in-plane.png

Let $\pr_{M, N}$ be the projection on $M$ along $N$:
 * $\forall x \in \R^2: \map {\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$.

Then $M$ is the set of fixed points of $\pr_{M, N}$ in the sense that:


 * $x \in M$


 * $\map {\pr_{M, N} } x = x$
 * $\map {\pr_{M, N} } x = x$

Sufficient Condition
Let $x \in M$.

Let $\LL$ be the straight line through $x$ which is parallel to $N$.

As $x \in M$ it follows that $x$ is on the intersection of $M$ with $\LL$.

Hence by definition:
 * $\map {\pr_{M, N} } x = x$

Necessary Condition
Again, let $\LL$ be the straight line through $x$ which is parallel to $N$.

Let $\map {\pr_{M, N} } x = x$.

Then by definition $x$ is on the intersection of $M$ with $\LL$.

Hence by definition of intersection:
 * $x \in M$.