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.
123
questions
0
votes
0
answers
18
views
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 ...
- 401
2
votes
1
answer
71
views
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
0
answers
48
views
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 ...
- 319
-4
votes
1
answer
67
views
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.
5
votes
0
answers
77
views
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 ...
- 23.5k
4
votes
1
answer
90
views
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 ...
- 1,349
3
votes
1
answer
99
views
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 ...
- 7,789
0
votes
0
answers
37
views
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 ...
0
votes
0
answers
51
views
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 ...
- 1
0
votes
1
answer
44
views
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_{...
- 322
0
votes
1
answer
309
views
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 ...
- 1,484
0
votes
1
answer
1k
views
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 ...
- 453
4
votes
2
answers
145
views
Flux integral of Gauss law
Consider a point charge enclosed by some surface, using spherical coordinates, and taking $\hat a$ to be the unit vector in the direction of the surface element, flux is
$$\oint\vec E\cdot d\vec A = ...
0
votes
0
answers
46
views
Solid Angles: 3D Polygon inside a sphere
My propositional logic about the topic is following and My question is the way to prove this:
If two or more points(vertex) of a 3D polygon share the same solid angle, then those points are on the ...
- 131
1
vote
1
answer
69
views
Spherical Means(average) with Taylor Expansion
I saw a formula in this paper A. D. Becke (1983). Hartree–Fock exchange energy of an inhomogeneous electron gas.
which is an integral about the spherical means:
$$
\frac{1}{4\pi} \int e^{\vec{s}\cdot\...
- 247