# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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, ...