I have ellipsoidal cap rotated in the x-y plane by an angle $\beta$; where the axis size in x coordinate is 'a', the axis size in y-coordinate is 'b' and axis size in z coordinate is 'c'. I am trying to figure out the projected area of the cap in the x-z plane from bounds of maximum y to maximum x and minimum y ( = - maximum y) to maximum x as shown for the elliptic base found from Page 15, 16 in and in Bounding box for rotated ellipse.
The points on an ellipsoid in spherical coordinates are given as $$x = (a \cos(\theta) \cos(\beta)-b \sin(\theta) \sin(\beta)) \sin(\phi)$$ $$y = (a \cos(\theta) \sin(\beta)+b \sin(\theta) \cos(\beta)) \sin(\phi)$$ $$z = c \cos(\phi)$$
Now to find the projected area of the cap in the x-z plane from bounds of maximum y to maximum x, I integrate projection of the an "infinitesimal surface area" on the ellipsoid $dA_{xz}$ from $\theta$ corresponding to the bounding box limits as derived above. This follows
$$A_{xz} = \iint dxdz = \iint det \frac{\partial(x,z)}{\partial(\theta,\phi)}d\theta d\phi = \int_{0}^{\phi_0}\int_{\theta_1}^{\theta_2} c(a\sin\theta\cos\beta+b\cos\theta\sin\beta)\sin^2\phi d\theta d\phi$$
where $\phi_0$ is the angle with z-axis at the cap height and the limits of integration are shown as the projected ellipse in x-y plane.
I use an ellipsoid with $a = 2, b = 1, c =1 $ rotated at an angle $\beta = 5 \pi /12$. I integrate from boundaries $ 0 $ to $ \phi_0 = 0.3176 $ and $ \theta_1 = 0.2302 $ to $ \theta_2 = 1.4422 $ to get and area $A_{xz}^1$ and from boundaries $ 0 $ to $ \phi_0 = 0.3176 $ and $ \theta_3 = \theta_2 - \pi $ to $ \theta_1 = 0.2302 $ to get and area $A_{xz}^2$. However, the area $A_{xz}^2$ is smaller than the area $A_{xz}^1$ even though $dA_{xz}^2$ has a larger difference in limits of integration than $dA_{xz}^1$, which is counter-intuitive. Is my expression for the integrand wrong? Please help.
Derivation of parametric angles of bounding box as limits of integration
I understand that $\theta$ is parametric for the point M and N as shown figure 1. I calculate parameters $\theta_1$ and by $\theta_2$ for $M$ and $N$ respectively for the rotated ellipse of axis size $a$ and $b$ along $x$ and $y$ axis respectively, as follows.
Step 1. Rotate back points $M$ and $N$ to get points $M^\beta$ and $N^\beta$ on the unrotated ellipse by multiplying coordinates of $M$ and $N$ with Rotation Matrix for $-\beta$ ($M_x$, $M_y$,$N_x$ and $N_y$ are x and y coordinates of the points of $M$ and $N$ as shown in as shown in figure 1)
$$ M_y^\beta = M_x\sin(-\beta) + M_y\cos(-\beta))$$ $$ N_y^\beta = N_x\sin(-\beta) + N_y\cos(-\beta))$$
Step 2. Scale back the rotated points on the ellipse to points on the parametric circle of radius $r$ as $ C{M_y^\beta}$ and $ C{N_y^\beta}$; for the ellipse: ellipticity ratio in x and y axis are $e_x =a/r$ and $e_y =b/r$ respectively
$$ C{M_y^\beta} = M_y^\beta/e_y $$ $$ C{N_y^\beta} = N_y^\beta/e_y $$
Step 3. Find the angle substended by the points on the circle with its center
$$ \theta^\beta_M = \arcsin{C{M_y^\beta}/r} $$ $$ \theta^\beta_N = \arcsin{C{N_y^\beta}/r} $$
Step 4. Rotate the angle again by $\beta$ to find the parametric angles corresponding to $M$ and $N$
$$\theta_1 = \theta^\beta_M + \beta $$ $$\theta_2 = \theta^\beta_N + \beta $$
Is the method correct?
Thanks.
Yes, your integrand is wrong. If you plug the values $\theta_1$ and $\theta_2$ into the formula $c(a\sin\theta\cos\beta+b\cos\theta\sin\beta),$ you will get points that are nowhere near the points where you measured $\theta_1$ and $\theta_2$.
The reason for this is that $\theta$ in $c(a\sin\theta\cos\beta+b\cos\theta\sin\beta),$ if we are to assign any geometric meaning to it, is the angle of a point on a circle before you distorted the circle to an ellipse and before you rotated the ellipse. The angle is changed both by the acts of turning the circle into an ellipse and by the rotation.
In your edited question, the derivation of $\theta_1$ and $\theta_2$ appears almost correct.
But the last step is incorrect. Just don't do it. The parametric values you want are $\theta_1 = \theta^\beta_M$ and $\theta_2 = \theta^\beta_N.$
Supposing that your calculations were correct up to the last step, if you then mistakenly add an offset of $\beta,$ it would explain why you get a positive value for $\theta_1$ when it should have been negative.
Try taking the values you get for $\theta_1$ and $\theta_2$ and use them as input to the formula of the ellipsoid to see what values of $(x,y,z)$ you get for each of those parametric values. Then compare those $(x,y,z)$ coordinates to the $(x,y)$ coordinates of $M$ and $N.$ If your calculations are all correct, the coordinates should match.