# Intersecting angle of point and line segment

Given a line segment $L$ in $\mathbb R^2$ ($2D$-Space), represented by two points, and points $P_1$ and $P_2$ (also in $\mathbb R^2$) not on the segment, $L$ rotates about $P_1$, its full rotation forming an annulus (doughnut-like shape). Assuming that $P_2$ is inside of the annulus, I need to find the angle that $L$ needs to rotate about $P_1$ so that it intersects $P_2$ (such that $P_2$ lies on $L$ after its rotation).

The closest to an answer I've seen is this. It uses 3 lines in $\mathbb R^3$ (with one rotating about another to intersect with the last line), and I can imagine my problem being an analogue to this, the $\mathbb R^3$ space being projected onto $\mathbb R_2$ with two of the lines perpendicular to the plane of projection, though I struggle to understand how to execute this (and I believe there must be a simpler solution). Nor can I find a simple trigonometric answer.

rbjacob

• Find the intersection of $L$ with the circle through $P_2$ with center $P_1$ and measure the resulting angle.
– amd
Sep 1, 2018 at 3:59

Just as @amd suggested:

• Calculate $r= |P_2 - P_1|$.
• Calculate $Q$, the intersection of the circle centered on $P_1$ of radius $r$, with $L$.
• Calculate the angle $\theta$ from $P_1 Q$ to $P_1 P_2$.

• Elegant solution, a lot simpler than I had conceived it would be. Thanks for the help Joseph and @amd!
– drj
Sep 2, 2018 at 4:03