Given R find F

The point F must be on the line through E and R. Thus $(F-E) \parallel (E-R)$ gives $(f-r,u,v) \parallel (r-c,-s,h)$ which in turn gives

$$\frac {f-r} {f-c} = \frac {u} {-s} = \frac {v} {h}$$ (4)

Thus,

$$(u,v) = \frac{f-r}{f-c}(-s,h).$$ (5)

which gives F.