# Perspective trigonometry

I was thinking of a question for a while now.

R1 is radius of small circle, R2 is radius of large. The small circle is actually the base of a right cylinder and the big circle is the top. So in reality R1=R2. Given that R2=k*R1, k is some proportion, in the picture is there a way to determine the height of the cylinder?

I came to this question because originally I was thinking if there is a way to figure out the distance from observer A to B with a camera. You essentially move the camera upwards perpendicular to ground by x meter at the same time detecting by how much observer B moved (just like the cylinder question distance B moved is simply how it looks like on 2D).

If $V$ is the viewpoint, then it must be aligned with the cylinder axis (dashed line in the diagram below), which in turn is perpendicular to the view plane $\pi$. If $v$ is the distance between $V$ and $\pi$ and $d$ is the distance between $\pi$ and the front base of the cylinder, then by similar triangles we have:
$$R_2:R=v:(v+d), \quad\hbox{and}\quad R_1:R=v:(v+d+h).$$ By dividing the first equation by the second we obtain: $$R_1:R_2=(v+d+h):(v+d).$$ To solve for $h$ you then need to know $d+v$.