# Questions tagged [solid-angle]

Analogue of radians on spheres. A sphere has solid angle $4\pi$ comparing to the $2\pi$ radian for a circle.

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### Understanding the Rendering equation geometry term

I am trying to understand a formulation of the rendering equation which includes the geometry term, denoted as $G(x,y)$ in the equation. I understand that $cos(N_i, \psi_i)$ is applying Lamberts ...
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### How can we show that $\int_S \frac{dS\cos\alpha}{r^2}=4\pi$ in spherical polar coordinates $(r,\theta,\phi)$?

To find the solid angle subtended at a point O by an arbitrary surface element $d{\vec S}=dS\hat{{n}}$, one joins the peripheral points of $d{\vec S}$ to O by straight lines which generates a cone at ...
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### Calculate the Solid Angle using Stokes' theorem

The solid angle for the surface S subtended at a point P is: $$\Omega=\iint_{S} \frac{\hat{r} \cdot \hat{n}}{r^{2}} d S$$ where $\hat{r}$ and $\hat{n}$ are unit vectors and $r =|\vec {r}|$ is the ...
1 vote
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### Solid angle with approximation and trigonometry$~ \omega_{} \approx \frac{ \pi a ^{2} \cdot \cos^{}\left(\theta_{} \right) }{ r ^{2} }$

I've drawn the below diagram. The circle has the radius $a$. $$\omega_{} \approx \frac{ \pi a ^{2} \cdot \cos^{}\left(\theta_{} \right) }{ r ^{2} } \tag{1}$$ $$a \ll r$$ I viewed diagrams ...
1 vote
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### Intuitive explanation of solid angles as a natural 3-dimensional analogue of angles

I'm searching for an intuitive explanation of solid angles as a natural 3-dimensional analogue of angles. It's not sound yet, but I would like to say that the length of the arc occupied by the ...
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### Is there a simple solution to this spherical n-dimensional geometry problem arising in probability setting?

Find the relative measure of the space defined by $$Z\cdot a \geq 0, \quad Z \cdot b \geq 0, \quad Z \cdot 1=0$$ to the unconstrained problem $$\quad Z \cdot 1 = 0$$ where $Z, a, b, 1 \in R^d$ and ...
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### Orienting a solid angle

I'm working on a project in which I need to somehow define oriented solid angle in Cartesian coordinate system, similar to how "regular" oriented angle is defined. And well, I have no idea how to do ...
1 vote