My goal is to calculate the shadow of a rotated rectangle in 3d space (with Y being the axis pointing upwards) for use in an equation for falling objects with air resistance (specifically for the projected area). The rotation is given as euler $x, y, z$ angles, and the shadow is the rhomboid that appears when the rotated rectangle is projected flat onto the XZ-axis.
The usecase for this is to calculate the projected surface area that affects air resistance when an object is falling downwards
I considered calculating the positions of the rotated vertices for a rectangle with dimensions $w \times h$ using matrix multiplication with the rotation matrix and constructing a flattened rhomboid, from which on the area can then be calculated. But doing this much trigonometry seems too computationally intensive for real-time usage, which made me wonder if an approximation for this could be made.
It's likely that I'm missing a simple way to approach this problem, since I'm rather new to rotations and matrices in general. Any input is appreciated.
First of all, construct once and for all your rotation matrix $R$ from Euler angles (I don't embark into that task because there are too many conventions around).
Read off then the cosine of the overall rotation angle, which is just the $y$ coordinate of the transformed of $(0,1,0)$ (the unit normal vector), i.e. it is element $R_{22}$ in your matrix.
The area of the shadow is then simply $A\cdot R_{22}$, where $A$ is the area of the rectangle.
EDIT.
Consider a small rectangle of area $\Delta A$, inside your rectangle, having a side parallel to the ground (i.e. to $xz$ plane). If $\theta$ is the rotation angle of the normal vector, the area of the shadow is then clearly $\Delta A\cdot\cos\theta$. But you can decompose your large rectangle into a sum of rectangles like that, hence...
Finally, note that $\cos\theta$ is the projection of the unit normal vector of the rotated rectangle onto the vertical axis ($y$ axis), that is: $\cos\theta=R_{22}$.