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$$ \cos(\theta_{max}) = \sqrt{1 - \frac{R^2}{length^2(\mathbf{c} - \mathbf{p})}} $$
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We also need to evaluate the PDF of directions. For a uniform distribution toward the sphere the PDF
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- is $1/\mathit {solid_angle}$. What is the solid angle of the sphere? It has something to do with the
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- $C$ above. It is -- by definition -- the area on the unit sphere, so the integral is
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+ is $1/\text {solid_angle}$. What is the solid angle of the sphere? It has something to do with
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+ the $C$ above. It is -- by definition -- the area on the unit sphere, so the integral is
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- $$ \mathit{solid angle } = \int_{0}^{2\pi} \int_{0}^{\theta_{max}} \sin(\theta)
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+ $$ \text{solid_angle } = \int_{0}^{2\pi} \int_{0}^{\theta_{max}} \sin(\theta)
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= 2 \pi \cdot (1-\cos(\theta_{max})) $$
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It’s good to check the math on all such calculations. I usually plug in the extreme cases (thank you
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for that concept, Mr. Horton -- my high school physics teacher). For a zero radius sphere
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- $\cos(\theta_{max}) = 0 $, and that works. For a sphere tangent at $\mathbf{p}$,
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+ $\cos(\theta_{max}) = 1 $, and that works. For a sphere tangent at $\mathbf{p}$,
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$\cos(\theta_{max}) = 0$, and $2\pi$ is the area of a hemisphere, so that works too.
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