Given F find R

Since the point R is on the cylinder, it has coordinates (c, s, h) where c² + s² = 1. R also lies on the line through E and F and so, for some t, R = E + t (F-E), which gives

(c,s,h) = (r + t ( f- r), t u, e + t v) (1)

Thus 1 = (c² + s²) = (r + t (f - r))² + (t u)². Expanding to powers of t, we get

(u² + (f-r)²) t² + 2 r (f-r) t + r² - 1 = 0 (2)

We wish to find the smallest (if any) of the roots of (2). Using the quadratic equation, we find

$$t = \frac {r (r - f) - \sqrt{(r-f)^2 - u^2(r^2-1)}} {u^2 + (f-r)^2}.$$ (3)

Substitution of this value of t into (1) gives the coordinates of R.