As a follow up to Average projected area in higher dimension, I started to think about a variation of the original post. Let's visualize a circle with radius $1$ in $3D$ space and cast its shadow on the $x$-$z$ plane. The area of the shadow will change as we orient the circle in space. After we obtain all possible orientations of the circle, what will the average area of the shadow be? Just for clarification, average area is the sum of projected area divided by the number of possible orientations. For this problem, there are infinitely many orientations, so we have to apply calculus here. I think we have to set an integral over the circle's area, but I am not sure how exactly I should do that. Any help or related information would be truly appreciated.
1 Answer
$\begingroup$The problem can be reformulated probabilistically:
Uniformly choose a random unit vector and take the unit circle perpendicular to that vector. What is the expected area of the circle's projection onto the $xy$-plane?
By symmetry, we can consider only random vectors with positive $z$-coordinate. Let the angle the chosen unit vector makes with the $xy$-plane be $\varphi$, then its pdf is $\cos\varphi$ for $0\le\varphi\le\pi/2$. At a fixed $\varphi$ the circle's projection is an ellipse with semi-axes 1 and $\sin\varphi$, thus area $\pi\sin\varphi$. Therefore the expected area of the projection is$$\int_0^{\pi/2}\pi\sin\varphi\cos\varphi\,d\varphi=\frac\pi2$$
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