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 ...
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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 ...
Vincent Topacio's user avatar
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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 ...
nick012000's user avatar
4 votes
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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 = ...
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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 ...
Seung Hwan Kim's user avatar
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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\...
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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 ...
Solidification's user avatar
1 vote
1 answer
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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 integral, with legendre polynomials

I've been trying to solve this question from Jon Mathews Mathematical methods for physics, and I'm honestly very lost, I was given the following hint: $$ \cos \gamma=\cos \theta \cos \theta^{\prime}+\...
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Calculate the area of the hemisphere cut by a plane

I have the following problem. There is a unit hemisphere cut by the plane passing through the diameter. The angle $\gamma$ is given. The plane cuts a half of the great circle. I need to find the area ...
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Solid angle of $M^2 \subset \mathbb{R}^3$, when $0 \in M^2$ and $0 \notin M^2$

Let $M^2 \subset \mathbb{R}^3$ be a manifold (surface, in this case), with regular boundary $\partial M$, such that $0 \notin \partial M$. In Do Carmo's book "Differential Forms and Applications&...
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Pyramid - Cartesian Space xyz

I have a pyramid (in general with a rectangular base) like the following: with: Angle: $\widehat{AVB} = 30°$ Angle: $\widehat{BVC} = 40°$ Edge $\overline{VO} = 100$. It is in the space $xyz$, with ...
VittorioC's user avatar
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Integrate over $g(\vec{v} \cdot \vec{x}) \ h(|x|)$ using solid angle and polar coordinates

Relating to a question about fourier transforms I want to solve the n-dimensional integral for $n > 2$ $$ \int_{x \in \mathbb{R}^n} f(x) = \int_{x \in \mathbb{R}^n} g(\vec{v} \cdot \vec{x}) \ h(|x|)...
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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 ...
electrical apprentice's user avatar
1 vote
1 answer
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How to calculate the solid angle spanned by 3 vectors? [duplicate]

I find an equation from wiki https://en.wikipedia.org/wiki/Solid_angle#cite_note-6, $$\tan{\frac{\Omega}{2}}=\frac{|\vec{a}\cdot(\vec{b}\times\vec{c})|}{|a||b||c|+(\vec{a}\cdot\vec{b})|c|+(\vec{a}\...
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Solid angle subtended by polar cap

Solid angle subtended by polar cap at unit sphere center latitude $\phi$ is $$ 2 \pi (1- \sin \phi_c)$$ What is the solid angle it subtends at other unsymmetric points inside the sphere like ...
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Addition formulas for direction cosines?

I will quickly explain what I am expecting from this question. Given a point $p=(x,y)$ in the plane, we can look at the ratios $$A_1(p)=\dfrac{x}{r},\qquad A_2(p)=\dfrac{y}{r},$$ where $r$ is the ...
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Solid angle of a pyramid

Suppose I have a rectangular pyramid. I partition the dihedral angle between a fixed pair of opposite faces into three parts and thereby obtain three sub-pyramids (within the original one). Consider ...
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Random walk on a sphere: statistics of the solid angle?

Consider a random path on the surface of a 2-sphere, made of N discrete points (each picked with a uniform distribution across the surface). The path connects two successive points by the shortest ...
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Interior angle of a polyhedral cone

What is the angle subtended by a polyhedral cone $\{\pmb{\theta}\in\mathbb{R}^{m}:A\pmb{\theta}\ge\pmb{0}\}$ at its vertex (the origin) where $A$ is a full-rank matrix ? Also what is the solid angle ...
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Finding the integral over planar angles

I'm having difficulty proving the LHS equals the RHS. Both the limit and integral is giving me problems. $$ 4\pi\,\mathrm{rad}^2 ≟ \lim_{n\to\infty}n\int_{-\pi/2}^{\pi/2} \cos^{-1}(\sin(φ)² + \cos(φ)² ...
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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 ...
Proki's user avatar
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Formula for the Volume of a Spherical Triangle given the Solid Angle

I'm working on a problem for which I have to calculate the volume of a spherical triangle given the Solid Angle Ω. Specifically, if the Solid Angle Ω (in steradians) span by 3 vectors $\...
Harris Armeniakos's user avatar
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Do the triangles in an "icosphere" (geodesic polyhedron) all have the same solid angle from the center?

An "icosphere" has the mathematical name geodesic polyhedron. It's an approximation to a sphere made out of triangles with either 5 or 6 triangles meeting at a vertex. It can be made by a subdivision ...
Ben Crossley's user avatar
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solid angles of an n-simplex

Do there exist formulae relating the n-th dimensional solid angles of an n-simplex to either the n-th order dihedral angles, the volume of the n-1 dimensional facets, or the side lengths of the ...
user208480's user avatar
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How to calculate a directionally averaged distribution?

I'm trying to work out how to find the directional average of a velocity distribution (where the input velocity is a 3d vector). It has been quoted as below: $$f(v) = \oint f(\textbf{v})d\Omega _v $$ ...
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Can one always interchange the order of a surface and volume integral?

Consider a continuous charge distribution in volume $V'$. Draw a closed surface $S$ inside the volume $V'$. Consider the following multiple integral: $$A=\iiint_{V'} \left[ \iint_S \dfrac{\...
Joe's user avatar
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Octahedral facet solid angle

I'm trying to get an equation for a solid angle of a segment of octahedron in the same vein as described in this article cubemap-texel-solid-angle. I ended up having to integrate $$\int \int \frac{1}{(...
Anton Schreiner's user avatar
1 vote
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Dividing a solid angle into equal parts

In 2D, using polar coordinates, I have divided a unit circle into n equal parts (of equal $\theta$) and been able to form equations for the radius arc between the $\theta$ boundaries of the m'th ...
Drex's user avatar
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What exactly is meant by convex surfaces?

Google search didn't show up. It just shows up information related to spherical mirrors everywhere. Is there a way to intuitively (and maybe formally) define convex (and concave) surfaces around a ...
Joe's user avatar
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4 votes
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Areas of tetrahedron faces in proportion to opposite solid angles?

Is there a relationship analogous to the law of sines for triangles, but for tetrahedra? A natural generalization would be $$ a : b : c : d \;=\; \sin A : \sin B : \sin C : \sin D $$ where $a,b,c,d$ ...
Joseph O'Rourke's user avatar
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Solid angle: Must a region subtending a solid angle be (simply) connected?

Although answers to the question "What is a Solid Angle?" explain that the shape of the area subtending a solid angle doesn't matter, my question is if the region has to be simply connected (no holes)....
FizzleDizzle's user avatar
1 vote
1 answer
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Question on Solid Angles and Linking Number

In Spivak's Calculus on Manifolds there is a question about solid angles and linking number that confuses me. Suppose $f([0,1])=\partial M$ where $f$ is a closed curve and $M$ is a compact oriented ...
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2 votes
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How to prove this identity involving dot product of solid angle and gradient

How to prove following for $n\geq0$. $$\int_{4\pi}d\vec{\Omega}(\vec{\Omega}\cdot\vec{\nabla})^{2n}f(\vec{r})=\frac{4\pi}{2n+1}\nabla^{2n}f(\vec{r})$$ Where, at any point $\vec{r}$, the $\vec{\Omega}$ ...
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How to calculate solid angle of nonspherical surface?

My objective is to calculate (or find an expression) for the solid angle of a circular loop (parametrized by $(x,y)=(r \cos{x_i}, r \cos{y_i})$, where $0\leq x, y \leq 2\pi$) on the surface defined as:...
TribalChief's user avatar
7 votes
2 answers
1k views

Can there be two adjacent solid angles?

Thanks for reading. My real question is the second part - in the first part I'm just explaining myself. Please read through! Thanks. In 2D geometry, it is easy to picture what it means to add up 2 ...
joshuaronis's user avatar
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1 vote
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Derivation of $\frac{\cos(\theta)dA}{r^2} = d\omega$

I've been looking for a (formal) derivation of the following equation $\frac{\cos(\theta)dA}{r^2} = d\omega$. Where $d\omega = \sin(\theta_x)d\theta d\phi$ is the differential solid angle, and $dA$ is ...
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Averaging a function over solid angle

I am trying to average $r$ over the solid angle $\Omega$ in 3D. To start this I have expressed $r$ in terms of the angle $a$ and sides $x$ and $d$ in 2D with the help of the law of cosines: $r = x*cos(...
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