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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Does such a formula for solid angle exist?
Let there be a cube of side $40$ units whose center is at the origin of Cartesian coordinate system.
Let:
$(a,b,c)$ be any point outside the cube
$s,t,u \in \{ -,+ \}$
$A_{(s,t,u)} = a\ (s)\ 20$
$B_{(...
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Finding the solid angle for a viewer in a movie theatre, from viewer to screen but at different heights...
I am doing a research paper to find the best seat in the cinema. I want to find the solid angle of all possible seats, so I searched for a formula to do so.
I came across someone asking a similar ...
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Solid Angle Subtended By a Rectangle [closed]
Consider a rectangle with edges $2a$ and $2b$ in the $xy$-plane with its centre (the point of intersection of diagonals) at the origin. The problem is to find a closed form for the solid angle ...
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Computing the solid angle of a arbitrary closed surface.
Wikipedia page states
An object's solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the apex, that the object covers.
It also states that
The solid angle, $...
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Prove that all three bisecting planes of the dihedral angles of a trihedral angle intersect along one straight line.
Prove that all three bisecting planes of the dihedral angles of a trihedral angle intersect along one straight line. I attempted like this we can take one triangle at first as all its bisectors will ...
user1278719
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What's the hypersolid angle of a 5-cell (4d tetrahedron)?
It's known that the solid angle of the vertex of a regular tetrahedron is $\arccos(\frac{23}{27})$, or equivalently, $\frac\pi2-3\arcsin(\frac13)$ or $3\arccos(\frac13)-\pi$. (Trig identities are ...
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Solid angle of human field of vision
This is a question about solid angles.
According to Wikipedia, the central/binocular field of human vision is about $2\pi/3$ in the horizontal plane, and $\pi/3$ in the vertical axis.
Roughly, this ...
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How to calculate the solid angle of a rectangle?
Let $R$ be a rectangle with vertices $\boldsymbol{n}_1$, $\boldsymbol{n}_2$, $\boldsymbol{n}_3$ and $\boldsymbol{n}_4 \in \mathbb{R}^3$. I am looking for a formula for calculating the solid angle ...
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Flux integral of Gauss law
Consider a point charge enclosed by some surface, using spherical coordinates, and taking $\hat a$ to be the unit vector in the direction of the surface element, flux is
$$\oint\vec E\cdot d\vec A = ...
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Spherical Means(average) with Taylor Expansion
I saw a formula in this paper A. D. Becke (1983). Hartree–Fock exchange energy of an inhomogeneous electron gas.
which is an integral about the spherical means:
$$
\frac{1}{4\pi} \int e^{\vec{s}\cdot\...
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Solid angle subtended by a 3D surface from the line integral along the edge (Stokes theorem)
The solid angle subtended by the surface S 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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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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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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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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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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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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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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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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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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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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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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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
$$\int \int \frac{1}{(...
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What exactly is meant by convex surfaces?
Google search didn't show up. It just shows up information related to spherical mirrors everywhere.
Is there a way to intuitively (and maybe formally) define convex (and concave) surfaces around a ...
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Solid angle: Must a region subtending a solid angle be (simply) connected?
Although answers to the question "What is a Solid Angle?" explain that the shape of the area subtending a solid angle doesn't matter, my question is if the region has to be simply connected (no holes)....
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How to prove this identity involving dot product of solid angle and gradient
How to prove following for $n\geq0$.
$$\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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