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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questions
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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
$$\int \int \frac{1}{(...
1
vote
0answers
42 views
Dividing a solid angle into equal parts
In 2D, using polar coordinates, I have divided a unit circle into n equal parts (of equal $\theta$) and been able to form equations for the radius arc between the $\theta$ boundaries of the m'th ...
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0answers
25 views
What exactly is meant by convex surfaces?
Google search didn't show up. It just shows up information related to spherical mirrors everywhere.
Is there a way to intuitively (and maybe formally) define convex (and concave) surfaces around a ...
3
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0answers
35 views
Areas of tetrahedron faces in proportion to opposite solid angles?
Is there a relationship analogous to the law of sines for triangles, but for tetrahedra? A natural generalization would be
$$
a : b : c : d \;=\; \sin A : \sin B : \sin C : \sin D
$$
where $a,b,c,d$ ...
0
votes
0answers
29 views
Solid angle: Must a region subtending a solid angle be (simply) connected?
Although answers to the question "What is a Solid Angle?" explain that the shape of the area subtending a solid angle doesn't matter, my question is if the region has to be simply connected (no holes)....
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1answer
31 views
Question on Solid Angles and Linking Number
In Spivak's Calculus on Manifolds there is a question about solid angles and linking number that confuses me. Suppose $f([0,1])=\partial M$ where $f$ is a closed curve and $M$ is a compact oriented ...
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52 views
How to prove this identity involving dot product of solid angle and gradient
How to prove following for $n\geq0$.
$$\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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vote
0answers
43 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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votes
2answers
136 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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0answers
18 views
Integrate orthogonal function over solid angle
How do I integrate a product of Legendre polynomials over a volume?
So I understand that bunch of complete basis orthogonal basis as well. i.e. for Legendre polynomial, $\int_{-1}^1P_n(x)P_m(x)dx=\...
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0answers
100 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
114 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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0answers
64 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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0answers
49 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,...
0
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1answer
226 views
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\...
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votes
0answers
34 views
Is there any kind of trigonometry analog for solids?
I'm looking for a general method of calculating angles in convex tetrahedra, a 3-dimensional analog of trigonometry. Have someone established such system in a formal way?
0
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1answer
74 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 ...
3
votes
0answers
63 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
56 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 = {...
0
votes
2answers
194 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
45 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
115 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
66 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
votes
0answers
105 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 ...
0
votes
0answers
106 views
Solid angle with Stokes theorem
Trying to evaluate a solid angle of a spherical cap with Stokes theorem:
$$
\begin{gathered}
\int_\Omega \frac{\hat r}{r^2}\cdot d\vec\omega\\
A = - \frac{\cot\theta}{r} \hat\phi\\
\nabla \times A = \...
5
votes
0answers
107 views
5 dimensional angles (not 2D angles in 5 dimensions)
Given $2$, $2D$ vectors we can calculate the angle inside these vectors using either dot or cross product. - (And presumably many other methods too)
Given $3$, $3D$ vectors, how would I calculate the ...
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vote
2answers
576 views
How do I compute the solid angle of a square in space in spherical coordinates?
I am trying to find out how to calculate a solid angle of a square or a rectangle in space, in a situation where
we know θ and ϕ, being θ the polar angle and ϕ the azimutal angle
the sphere has ...
0
votes
1answer
425 views
Solid angle in a cube
Consider this picture.
I need to integrate a function of spatial position and direction for the entire volume of the cube. So for example at point (x1,y1,z1) I need to integrate for the distance s and ...
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vote
0answers
220 views
Directional integration involving delta functions and products thereof
I struggle to understand how integration over direction takes place in this paper (open access). I will give two examples. The first one is this:
$$C\int \int δ(1-s\cdot s')L(r,s')\,ds' = 2πCL(r,s)$$
...
0
votes
1answer
114 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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votes
1answer
40 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' ...
3
votes
0answers
91 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
56 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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0answers
83 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
votes
1answer
187 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
625 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, ...
7
votes
1answer
218 views
When do $n+2$ points in $\mathbb{R}^n$ lie on a same $(n-1)$-sphere?
When $n=2$, the following results are well-known:
Proposition 1. Let $A,B,C,D$ be $4$ distinct points in $\mathbb{R}^2$. They are aligned or cocyclic if and only if: $$\left(\overrightarrow{CA},\...
-2
votes
1answer
95 views
how can i integrate this integrand?(solid angle of circular loop) [closed]
hi im solving some electrodynamics problem but im troubled with integration.
i don`t know how to integrate this integrand analytically.
what i want to integrate is ...
8
votes
1answer
465 views
Can the inscribed angle theorem be generalized to solid angles in 3D? And beyond to n-dimensional space?
The "inscribed angle theorem" is a common 2-dimensional plane geometry fact. It states that for a circle the angle formed between any two points on the circumference with the center is twice the angle ...
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0answers
200 views
Division of Solid Angle When Subdividing Spherical Triangle
Suppose I have a spherical triangle (no special properties; in particular, not equilateral) with a known solid angle. Now, I divide it into four new spherical triangles by bisecting each edge:
...
0
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1answer
141 views
How to differentiate cos(x) where x is in steradians?
Its been long time I did some differentiation, nevertheless I went through the basic differentiation of all trigonometric functions but couldn't understand why the differentiation of $cos(x)$ is $2 \...
3
votes
1answer
764 views
Why is $|\cos\theta d\omega|$ the projection of the differential solid angle $d\omega$ onto the $(x,y)$-plane?
Let $B\subseteq\mathbb R^3$ be the ball with radius $r>0$ around $0$ and $S_{\partial B}$ be the surface measure of the boundary $\partial B$. Given a piece of the surface $A\subseteq\partial B$, ...
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1answer
281 views
Integral over a solid angle
I've been reading about energy conservation and radiosity from the perspective of computer graphics. The basic idea is simple enough: For all possible incoming light directions $\vec{l}$ and view ...
2
votes
1answer
792 views
Given a solid angle, how do I calculate the surface area subtended by the angle on the surface of a sphere?
I am interested in calculating the elemental surfaces $dS$ (AF) and d$S'$ (EG), given that the solid angle of the very tiny cones' apex, $H$ are both dΩ. Knowing this will help me prove Newton's ...
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0answers
883 views
Calculate radius and angle of circle connecting two vectors
I have two vectors that lie on a circle. How do I calculate the radius of the circle and the angle between the two lines from the center of the circle to the two vectors?
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70 views
Ulam Spiral, what angle does x fall on?
Morning all,
I'm trying to work out what angle a given number will fall on within the Ulam Spiral.
The formula I have so far is this:
$$ \dfrac{180 \times\sqrt{x}-255}{360} $$
For example using $x= ...
4
votes
1answer
3k views
How to calculate a solid angle (in Steradians) given only Horizontal Beam angle and Vertical Beam angle data.
I would like to convert a rectangular beam shape given in Horizontal and Vertical beam angle, into solid angle representing the surface area in steradians of projected light. For example a light ...
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vote
1answer
140 views
How to find out the solid angle subtended by this slotted section?
I have got to calculate total luminous flux incident on a plane section by calculating the solid angle subtended by the slotted (plane) section, having four identical rectangular slots each having ...
0
votes
1answer
480 views
How to evaluate solid angle subtended by a segmented circle?
The diagram above shows a circular plane, centered at the origin 'O', has a radius $7 cm$. Two identical rectangular strips, each having width $2 cm$, are thoroughly cut off from the circular plane ...
