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 ...
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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 ...
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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 ...
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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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Discrete solid angle

I'm trying to calculate the solid angle at a vertex. In my research, I found that the solid angle is equivalente to the angle excess on unit sphere projection. However, when I tested this formula on ...
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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 ...
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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 $\...
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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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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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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{\...
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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}{(...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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