# 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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### What is the substantial generalization of the “acuteness” to 3-simplex?

A 2-dimensional simplex is just a triangle, and a 3-dimensional simplex is just a tetrahedron… therefore for convenience, I simply use the term triangle and tetrahedron in the following words. We know ...
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### Prove that all three bisecting planes of the dihedral angles of a trihedral angle intersect along one straight line.

Prove that all three bisecting planes of the dihedral angles of a trihedral angle intersect along one straight line. I attempted like this we can take one triangle at first as all its bisectors will ...
1 vote
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### Clarification Regarding Solid Angle

I am studying Zangwill's Modern Electrodynamics but I'm having trouble following an argument he makes about solid angles in preparation for deriving the integral form of Gauss's law. He defines the ...
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### Arnold's Trivium problem 69

Does anyone have solution for Arnold's Trivium problem #69? Prove that the solid angle based on a given closed contour is a function of the vertex of the angle that is harmonic outside of the contour.
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### What's the hypersolid angle of a 5-cell (4d tetrahedron)?

It's known that the solid angle of the vertex of a regular tetrahedron is $\arccos(\frac{23}{27})$, or equivalently, $\frac\pi2-3\arcsin(\frac13)$ or $3\arccos(\frac13)-\pi$. (Trig identities are ...
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### Reference Request for Solid Angles

I'm looking for a reference that has a discussion of solid angles. Many facts about them are available in various places online, but I haven't had any luck finding a text that treats them. I might be ...
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### Solid angle of human field of vision

This is a question about solid angles. According to Wikipedia, the central/binocular field of human vision is about $2\pi/3$ in the horizontal plane, and $\pi/3$ in the vertical axis. Roughly, this ...
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### What is the solid angle substended by a point on some closed surface?

I'm trying to find out the solid angle subtended over the entirety of some closed surface S by some point P located on the surface. For a point within the surface, the answer is of course 4$\pi$, but ...
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### Fraction of Solid Angle

I wonder which way is the correct way to calculate a specific fraction of a solid angle. I divided a hemisphere into a number of solid angles by using weights of gauss quadrature in the zenith ...
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### What is the asymptotic version of the solid angle formula in $d$ dimensions?

It is well known that the solid angle in an euclidian space of $d$ dimensions ($d = 2 n$ or $d = 2 n + 1$, where $n = 1, 2, 3, \dots, \infty$) is given by these formulae: \begin{align}\tag{1} \Omega_{...
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### How to calculate the solid angle of a rectangle?

Let $R$ be a rectangle with vertices $\boldsymbol{n}_1$, $\boldsymbol{n}_2$, $\boldsymbol{n}_3$ and $\boldsymbol{n}_4 \in \mathbb{R}^3$. I am looking for a formula for calculating the solid angle ...
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### How to calculate solid angles of segments of hemisphere from abstract points

I am imagining a half-sphere which has been cut, pizza-style, into many slices, which may vary in size. I want to specify a point inside the semi-sphere at random and be able to identify what solid ...
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### Does a sphere really have an area of 41,000 square degrees? [closed]

So, after reading the latest XKCD comic and it accompanying page on the explainxkcd wiki, I saw a link to this site that claims that a sphere has a surface area of approximately 41000 square degrees ...
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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 ...
1 vote
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### Solid angle subtended by a 3D surface from the line integral along the edge (Stokes theorem)

The solid angle subtended by the surface S 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 ...
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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 ...
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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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