Projecting a cone on a surface

Question

I want to find the projection of a base of a cone on a plane. I'm looking for its area. I know only the height of the cone and two angles and the distance from the vertex to the plane. Could you give me some pointers?

Answer 1

Let's apply the sine rule to the upper triangle on the right: if $p_1$ is its horizontal (blue) side we have $$ {p_1\over\sin(\pi/2+\Omega)}={R\tan\Omega\over\sin(\pi/2-\Omega-\alpha)}, \quad\hbox{whence:}\quad p_1=R\tan\Omega{\cos\Omega\over\cos(\alpha+\Omega)}. $$ In an analogous way, by applying the sine rule to the upper triangle on the left, its horizontal (blue) side turns out to be: $$ p_2=R\tan\Omega{\cos\Omega\over\cos(\alpha-\Omega)}. $$ The major axis $2a$ of the ellipse is then $2a=p_1+p_2$.

To find semi-major axis $b$ we can substitute $x=(p_1-p_2)/2$ and $y=R\tan\Omega$ into the equation $$ {x^2\over a^2}+{y^2\over b^2}=1, $$ which gives: $$ b={R\tan\Omega(p_1+p_2)\over2\sqrt{p_1p_2}}. $$

From $a$ and $b$ you can compute the area of the ellipse.