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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2answers
123 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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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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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 ...
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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 $a,b,c,d$ ...
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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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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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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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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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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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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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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,...
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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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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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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?
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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 ...
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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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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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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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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 ...
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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 ...
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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 ...
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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 = \...
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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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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 ...
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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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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: $$C\int \int δ(1-s\cdot s')L(r,s')\,ds' = 2πCL(r,s)$$ ...
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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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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' ...
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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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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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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 ...
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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, ...
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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, ...
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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},\...
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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 ...
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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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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: ...
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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 \...
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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 ...
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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= ...
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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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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 ...
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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 ...