Mathematics Stack Exchange
Mathematics Stack Exchange

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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Why does the differential solid angle have a $\sin\theta$ term in integration in spherical coordinates?

When you integrate in spherical coordinates, the differential element isn't just $ \mathrm d\theta \,\mathrm d\phi $. No. It's $\sin\theta \,\mathrm d\theta \,\mathrm d\phi$, where $\theta$ is the inclination angle and $\phi$ is the azimuthal…
bobobobo
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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 formed by those two points with any other point…
user669487
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Solid angle: integration

Can somebody explain the equivalence between integrating over the surface of a unit sphere and integrating over solid angle? I have been trying to understand the following transformation using a Jacobian but have been unsuccessful: $$\iiint dr\…
Sam Manzer
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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 angles. For example, in this random picture I got…
joshuaronis
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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:…
C. Falcon
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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 intersect at one common point which can be proved…
user1278719
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Solid angle and projections

During my studying of physics, I've been introduced to a concept of a solid angle. I think that I do understand it pretty good, however, I'm stuck with one certain problem. We know that a solid angle is $S/r^2$ where $S$ is the area subtended by a…
stew281
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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 $\sin\theta\,{\rm d}\theta {\rm d}\phi = {\rm d}\cos\theta{\rm d}\phi$
DJA
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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 weird.) This is around $0.551$ steradians, or around…
Akiva Weinberger
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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 solid angle inside them? Given $4$, $4D$ vectors,…
Ben Crossley
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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 projection that has a 120° H beam by 5° Vertical beam…
luminary
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What is the solid angle at a vertex in a snub cube?

I was able to find or derive an expression for most other Platonic or Archimedean solids, but for the snub cube I was not able to find a value nor find an expression anywhere.
nvcleemp
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Online Math Open Contest 2 Problem 50

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be the largest possible value of the circumradius…
Ahaan S. Rungta
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Reference Request for Solid Angles

I'm looking for a reference that has a discussion of solid angles. Many facts about them are available in various places online, but I haven't had any luck finding a text that treats them. I might be barking up the wrong tree, but neither Do Carmo's…
kandb
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A tricky integral (flux of a point charge through a disk)

The integrals: $$ \oint \frac{r\,dr\,d\phi}{\left(L^2+r^2+h^2+2Lr\cos\phi\right)^{3/2}}\\ \oint \frac{dx\,dy}{\left((L+x)^2+y^2+h^2\right)^{3/2}} $$ If we have a point charge at the origin and we want to find the flux through a disk of radius $R$…
iman
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