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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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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1answer
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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26 views
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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1answer
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.
$$ 4\pi\,\mathrm{rad}^2 ≟ \lim_{n\to\infty}n\int_{-\pi/2}^{\pi/2} \cos^{-1}(\sin(φ)² + \cos(φ)² ...
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12 views
Density in solid angle $\Omega$ convert to spherical angles $\theta$ and $\phi$
I was given a probability density $$ \frac{dP}{d\Omega} $$ where $\Omega$ is the solid angle such that $$ d\Omega = \sin \theta \ d\theta \ d\phi $$ and $\theta$ and $\phi$ the sperical coordinates ...
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1answer
33 views
Intuitive explanation of solid angles as a natural 3-dimensional analogue of angles
I'm searching for an intuitive explanation of solid angles as a natural 3-dimensional analogue of angles.
It's not sound yet, but I would like to say that the length of the arc occupied by the ...
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1answer
48 views
Is there a simple solution to this spherical n-dimensional geometry problem arising in probability setting?
Find the relative measure of the space defined by
$$
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 ...
0
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1answer
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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1answer
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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46 views
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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1answer
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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27 views
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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22 views
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 $$
...
2
votes
1answer
149 views
Can one always interchange the order of a surface and volume integral?
Consider a continuous charge distribution in volume $V'$. Draw a closed surface $S$ inside the volume $V'$.
Consider the following multiple integral:
$$A=\iiint_{V'} \left[ \iint_S \dfrac{\...
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votes
2answers
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
$$\int \int \frac{1}{(...
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0answers
303 views
Dividing a solid angle into equal parts
In 2D, using polar coordinates, I have divided a unit circle into n equal parts (of equal $\theta$) and been able to form equations for the radius arc between the $\theta$ boundaries of the m'th ...
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0answers
27 views
What exactly is meant by convex surfaces?
Google search didn't show up. It just shows up information related to spherical mirrors everywhere.
Is there a way to intuitively (and maybe formally) define convex (and concave) surfaces around a ...
3
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0answers
46 views
Areas of tetrahedron faces in proportion to opposite solid angles?
Is there a relationship analogous to the law of sines for triangles, but for tetrahedra? A natural generalization would be
$$
a : b : c : d \;=\; \sin A : \sin B : \sin C : \sin D
$$
where $a,b,c,d$ ...
1
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0answers
36 views
Solid angle: Must a region subtending a solid angle be (simply) connected?
Although answers to the question "What is a Solid Angle?" explain that the shape of the area subtending a solid angle doesn't matter, my question is if the region has to be simply connected (no holes)....
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1answer
52 views
Question on Solid Angles and Linking Number
In Spivak's Calculus on Manifolds there is a question about solid angles and linking number that confuses me. Suppose $f([0,1])=\partial M$ where $f$ is a closed curve and $M$ is a compact oriented ...
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80 views
How to prove this identity involving dot product of solid angle and gradient
How to prove following for $n\geq0$.
$$\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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71 views
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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2answers
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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1answer
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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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2answers
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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\...
0
votes
1answer
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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0answers
78 views
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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votes
2answers
84 views
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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2answers
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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1answer
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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0answers
164 views
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 ...
0
votes
1answer
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 ...
0
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0answers
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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0answers
136 views
Solid angle with Stokes theorem
Trying to evaluate a solid angle of a spherical cap with Stokes theorem:
$$
\begin{gathered}
\int_\Omega \frac{\hat r}{r^2}\cdot d\vec\omega\\
A = - \frac{\cot\theta}{r} \hat\phi\\
\nabla \times A = \...
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votes
0answers
143 views
5 dimensional angles (not 2D angles in 5 dimensions)
Given $2$, $2D$ vectors we can calculate the angle inside these vectors using either dot or cross product. - (And presumably many other methods too)
Given $3$, $3D$ vectors, how would I calculate the ...
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2answers
1k views
How do I compute the solid angle of a square in space in spherical coordinates?
I am trying to find out how to calculate a solid angle of a square or a rectangle in space, in a situation where
we know θ and ϕ, being θ the polar angle and ϕ the azimutal angle
the sphere has ...
0
votes
1answer
486 views
Solid angle in a cube
Consider this picture.
I need to integrate a function of spatial position and direction for the entire volume of the cube. So for example at point (x1,y1,z1) I need to integrate for the distance s and ...
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vote
0answers
272 views
Directional integration involving delta functions and products thereof
I struggle to understand how integration over direction takes place in this paper (open access). I will give two examples. The first one is this:
$$C\int \int δ(1-s\cdot s')L(r,s')\,ds' = 2πCL(r,s)$$
...
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1answer
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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1answer
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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votes
0answers
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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0answers
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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votes
0answers
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 ...
0
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1answer
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, ...
2
votes
1answer
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, ...
