I want to extend the following to work on a solid angle:

Suppose we have a volume filled with small surfaces. If we cast a ray from a given point, the probability that the ray will not hit a surface (i.e. is visible to the sky) is given as

$P(ray\ does\ not\ hit) = {e}^{-\alpha d/\cos\theta}$

where $\alpha$ is some decay factor, $d/cos\theta$ is the path length of the ray within the volume.

My question:
How can we compute the expected visibility of a ray if we consider the directions $(\theta, \phi)$, where $0<\theta<\pi/2$ and $0<\phi<2\pi$?

  • $\begingroup$ What do you mean by "consider the directions"? Are you trying to integrate over some solid angle instead of a single ray? Are you trying to extend the volume to be heterogeneous or anisotropic? $\endgroup$
    – Dan Hulme
    Dec 18, 2017 at 12:11
  • $\begingroup$ I am integrating over some solid angle. $\endgroup$
    – Stackmm
    Dec 18, 2017 at 19:59

1 Answer 1


You need to perform integration of $P$ over the hemisphere to calculate the solution. There doesn't seem to be closed-form solution as the solution requires incomplete gamma function: $$2\pi(e^{-\alpha d}-\alpha d\Gamma(0,\alpha d))$$


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