Can you please help me with the problem of computing the lateral area and volume of
a frustrum but the base and top ellipses are arbitrary 2 ellipses.
My problem is: if a1,b1 are the semi- major and minor axes of the base ellipse, the a2, and b2 are semi- major and minor axes of the top ellipse, a2 can be the minor semi-axis of the top ellipse, and b2 can be the major semi-axis, i.e., the major axes at the bottom and top can be orhogonal.
I have found obviously the solutions everywhere on the web, also here,
for the case where the base and top ellipses are similar in shape (that is the top ellipse is just a scaled ellipse of the base ellipse). So the Thales ratios between a1,a2,b1,b2 with height H are all known. Easy to compute the frustrum lateral area and volume as the difference of two elliptical cones. But here this does not work.
Is there analytical solution to this?
Thank you so much for reading. You are my last hope.
Best regards,
Zuheyr
Best Answer
A point $P_1$ on the base ellipse with azimuth $\theta$ has coordinates: $$ P_1=\left( {b_1\cos\theta\over\sqrt{1-e_1^2\cos^2\theta}}, {b_1\sin\theta\over\sqrt{1-e_1^2\cos^2\theta}}, 0 \right) $$ and the corresponding point $P_2$ on the top ellipse is then: $$ P_2=\left( {b_2\cos\theta\over\sqrt{1-e_2^2\cos^2\theta}}, {b_2\sin\theta\over\sqrt{1-e_2^2\cos^2\theta}}, H \right) $$ where $e_1$ and $e_2$ are the eccentricities: $$ e_1^2=1-{b_1^2\over a_1^2},\quad e_2^2=1-{b_2^2\over a_2^2}. $$ A generic point on segment $P_1P_2$ can be expressed as $$P=(1-t)\cdot P_1+t\cdot P_2, \quad\text{with}\quad t\in[0,1]. $$ This is then a parametrization of the lateral surface and the standard way to compute its area is: $$ area=\int_0^{2\pi}\int_0^1 \left|{\partial P\over\partial t}\times {\partial P\over\partial \theta}\right|\,dtd\theta. $$ For the volume, integrating in cylindrical coordinates gives: $$ volume={H\over2}\int_0^{2\pi}\int_0^1 \left(P_x^2+P_y^2\right)\,dtd\theta. $$