2D Coordinate - Given two lines and their angles, compute the point in which the perpendicular foot length becomes h

Question

I am given values of $x_1, y_1, x_2, y_2$, $\theta_{1}$, $\theta_{2}$, and $h$.

I want to find out the coordinate of the Unknown Point of Interest.

The Unknown Point of Interest is the point in which the length of the perpendicular foot between the two blue lines with $\theta_{1}$ becomes $h$. How can I do this?

Answer 1

Use trigonometry. If you know $h$ and the angle $\theta_1$, then you can treat the triangle as a right angled triangle and the length of the hypotenuse as $\frac{h}{\sin\theta_1}=h_1$. Then the point of interest can be written as $\{h_1\cos(\theta_1+\theta_2),h_1\sin(\theta_1+\theta_2)\}$ using the same trigonometric argument/transformation from polar co-ordinates to Cartesian.