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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54 views

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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50 views

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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11 views

Double solid angle integral for potential

I am stuck trying to integrate the following that arises in potential theory: $$f(v_1, v_2) = \int d^2\Omega_1 \int d^2\Omega_2 \frac{1}{|\mathbf{v}_1+\mathbf{v}_2+\hat{\mathbf{z}}|}$$ Here, $\hat{\...
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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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32 views

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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Density in solid angle $\Omega$ convert to spherical angles $\theta$ and $\phi$

I was given a probability density $$ \frac{dP}{d\Omega} $$ where $\Omega$ is the solid angle such that $$ d\Omega = \sin \theta \ d\theta \ d\phi $$ and $\theta$ and $\phi$ the sperical coordinates ...
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33 views

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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48 views

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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19 views

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 ...
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156 views

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 $\...
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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 ...
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100 views

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 ...
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If $p$ is uniformly sampled from an open rectangle $R\subseteq\mathbb R^3$, what's the distribution of the direction from $0$ to $p$ wrt solid angle?

Say we uniformly sample a point $p$ on a rectangle $R=(-a,a)\times(-b,b)\times\{1\}$, $a,b>0$, with surface area $A$. The density of $p$ with respect to the Lebesgue measure $\lambda^3$ on $\...
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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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149 views

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{\...
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221 views

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}{(...
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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 ...
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27 views

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 ...
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46 views

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 $a,b,c,d$ ...
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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)....
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52 views

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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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:...
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454 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 ...
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239 views

Solid angle subtended by an ellipse

This question introduces a formula for the evaluation of the solid angle corresponding to an ellipsoid. However, it is an approximation. Instead, this paper represents a general surface formula for ...
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1answer
168 views

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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173 views

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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102 views

Solid angle created from irregular polygon (over a sphere)

I have an $n$-polygon on a sphere ($n\geqslant3$). In this example the vertices are $C,D,E,F,G,H,I,J,K$. Which solid angle alpha generate this polygon respect origin of the sphere? For $C,D,E,F,G,H,I,...
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volume inside a solid solid angle

Is it right to say that in a sphere of radius R, the volume inside a solid angle $\Omega$ is just : $V=\frac{4\pi R^3}{3}\frac{\Omega}{4\pi}=\frac{R^3 \Omega}{3}$ ?
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How to show $\unicode{x222F} \dfrac{\hat{r} \times \vec{dS}}{r^2}=0$

I can do the following derivation using solid angle: $$\unicode{x222F} \dfrac{\hat{r} \cdot \vec{dS}}{r^2} =\unicode{x222F} \dfrac{dS \cos\alpha}{r^2} =\int^{2\pi}_0 \int^\pi_0 \sin\theta\ d\theta\ d\...
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94 views

Veach's thesis, projected solid angle sanity check

Here's an equivalence from Veach's thesis on light transport (page 88, 3.16): $$|\cos(\theta)|\sin(\theta)d(\theta)d(\phi) \equiv \\ \sin(\theta)d(\sin(\theta))d(\phi)$$ This seems wrong in the ...
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Is there an equivalent to trigonometry for solid angles?

Intuitively I would say that it would make no sense, but this question crossed my mind : Is there an equivalent to trigonometry for solid angles?? I haven't found anything yet. Thanks for answering!...
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Solid angle relation between sinθ dϕdθ and d(cos(θ))dϕ

I am a bit confused with regards to the concept of solid angle. Why is the solid angle which is defined as $\sin \theta {\rm d}\phi\, {d\rm }\theta$ equal to $\sin\theta\,{\rm d}\theta {\rm d}\phi = {...
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503 views

Calculating solid angle…

Here is a sketch of the problem statement : A cube of edge length $l$ is placed in three dimensional space with one vertex at the origin ${(0,0,0)}$ and all the faces parallel to the (Cartesian) ...
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81 views

What does it mean that a normal is inside a solid angle?

I'm reading through some stuff. And there are stuff like this mentioned. Assume we have some surface $\mathcal{S}$ we are focusing on some small element $d\mathcal{S}$ let's define $D(n)$ as the ...
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Is there a reason why a differential solid angle is represented by a cone?

Just wondering... is there a geometric reason why a differential solid angle is usually represented by a cone? I cannot see that given the formula $$ d\Omega = \sin \theta \, d\theta \, d\phi $$ I ...
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85 views

How do I get the exit angle of a body?

I have a 3D environment with vectors $(x, y, z)$. For example: Room size $10 \times 10 \times 10$ Bulb in the position $(3,5,10)$ Measuring points: $(5,5,0), (1,1,0), (5, 0, 5)$, etc. A light bulb ...
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163 views

Surface of a revolving probability density function given in spherical parameterization.

I wonder why the equation $(2)$ for steradians of a probability density function given in Henyey-Greenstein-Phase function is still dependent on the function of radius $p(\theta)$. So that the solid ...
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136 views

Solid angle with Stokes theorem

Trying to evaluate a solid angle of a spherical cap with Stokes theorem: $$ \begin{gathered} \int_\Omega \frac{\hat r}{r^2}\cdot d\vec\omega\\ A = - \frac{\cot\theta}{r} \hat\phi\\ \nabla \times A = \...
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5 dimensional angles (not 2D angles in 5 dimensions)

Given $2$, $2D$ vectors we can calculate the angle inside these vectors using either dot or cross product. - (And presumably many other methods too) Given $3$, $3D$ vectors, how would I calculate the ...
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How do I compute the solid angle of a square in space in spherical coordinates?

I am trying to find out how to calculate a solid angle of a square or a rectangle in space, in a situation where we know θ and ϕ, being θ the polar angle and ϕ the azimutal angle the sphere has ...
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486 views

Solid angle in a cube

Consider this picture. I need to integrate a function of spatial position and direction for the entire volume of the cube. So for example at point (x1,y1,z1) I need to integrate for the distance s and ...
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272 views

Directional integration involving delta functions and products thereof

I struggle to understand how integration over direction takes place in this paper (open access). I will give two examples. The first one is this: $$C\int \int δ(1-s\cdot s')L(r,s')\,ds' = 2πCL(r,s)$$ ...
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147 views

Integrating δ(1 - cosθ) over the entire solid angle in spherical coordinates.

I don't understand why the following is wrong: link here Should dθ sinθ be converted to -dcosθ? If yes should dθ sinθ be converted to -dcosθ for every integral involving cosθ?
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77 views

Integrating a function of φ and cosθ over the entire solid angle in spherical coordinates.

I found this integral in a book: link here The integrated function is a phase function or just a probability density function. I don't understand why instead of dω' = sinθ dφ dθ in this case dω' ...
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94 views

What proportions make a regular right prism a fair dice?

If the base of a right prism is a regular $n$-gon of side 1, what height makes it a fair dice? The $n=4$ case is obvious by symmetry. Assume constant density, constant downwards gravity, throwing on a ...
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63 views

rate of change of the angle of slant height from vertex point of a cone

I wish to find the rate of change of the angle of slant from the vertex of the cone when a particle is moving on a circular periphery(base of the cone) with an angular velocity of 'w'. I did this (...
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88 views

Angles of a Tetrahedron

Consider points A, B, C and D such that ∠ACB = 30◦, ∠CBD = 26◦, ∠DBA = 51◦ and ∠DAC = 13◦. Compute all possible values of the ∠BDC. This is a 3D-Geometry problem. So I am assuming ABCD is a ...
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248 views

Integrating Over a Product of (Non-Separable) Piecewise Functions (Hyper-Solid Angle of a Convex Polyhedral Cone)

My problem is as follows: given a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ where $n$ is some integer of order 10 and $f$ is defined by a product of (non-separable) linear piecewise functions, ...
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1k views

Integral of solid angle of closed surface from the exterior

Jackson derives Gauss's Law for electrostatics by transforming the surface integral of the electric field due to a single point charge over a closed surface into the integral of the solid angle, ...