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Solid angle in $D$ dimensions

Consider $d\Omega_D$, the element of solid angle in dimension $D$. Suppose an integrand depends only on $m-1$ of the angles (so that it doesn't depend on the set $\left\{\phi_i, i=1,\dots,D-m\right\}$ ...
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1answer
112 views

How to find out percentage of total area covered by 20 identical circles touching one another on a sphere?

20 identical circles touching one anther at total 30 different points (i.e. each one exactly touches three other circles) on a spherical surface with a finite radius. Thus all 20 circles are packed on ...
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0answers
55 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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46 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
146 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
46 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
71 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 ...
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0answers
41 views

How to calculate the solid angle subtended by the torus?

The diagram above shows a torus, with circular cross section, having inner radius $r=19 cm\space$ & outer radius $R=33 cm\space$. How to evaluate the solid angle subtended by the torus at a ...
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1answer
36 views

How to evaluate solid angle subtended by composite plane figure?

How to calculate the solid angle subtended by the shaded portion of the circular plane at a point $P(0,0,h)$. This composite plane figure is obtained by removing the right $\Delta ABC$ $(AB=BC)$ ...
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1answer
110 views

How to calculate solid angle of a rectangular detector of 20cm x 10cm?

I have an detector of $20cm*10cm$, how can i calculate the solid angle subtended by the detector if the detector is placed at $30$cm apart? Because directly i cannot use the formula ...
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1answer
19 views

Finding the formula of a line when given a line and angle.

Assume I have a line $y_1 = m_1x + b_1$. And I'm given $\theta$ degrees. How do I find a line $\theta$ degrees counterclockwise from the line.
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1answer
69 views

How to find out the solid angle subtended by a tetrahedron at its vertex?

Given that the angles between the consecutive lateral edges AB, AC & AD meeting at the vertex A of a tetrahedron ABCD are $ α, β, γ$ (as shown in the diagram below). Is there any set-formula to ...
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2answers
135 views

Why solid angle is defined as “the two-dimensional angle in three-dimensional space”?

In wikipedia the solid angle is defined as follows: In geometry, a solid angle (symbol: Ω) is the two-dimensional angle in three-dimensional space that an object subtends at a point. Why solid ...
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1answer
39 views

coordinates of 3rd point (vertex) of a right triangle knowing lengths and direction

In a last post I wanted to know the 3rd point of vertex, actually I have some similar problem .... I think I have all data.. for example...: 3 vertex cordinates in order to have the direction(gray ...
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2answers
187 views

Finding an angle between two vectors without a calculator

I have an exercise where I have to find the angle between 2 vectors without using the calculator. I know the formula which gives you the cosine of the angle between the 2 vectors, but I don't know how ...
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0answers
14 views

Locating point using 4 point data

currently I have 4 points of data on a circle, each point at a delta of 90 degrees. Like so: Now using this data I would like to locate point(s) on the circle depending on the size of the numbers. For ...
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1answer
32 views

Trapped volume of (n-1)-sphere inside an n-simplex

Assume that we take an $n$-simplex (with side length of 2 units) and place a unit $(n-1)$-sphere at each vertex. For $n=2$, half of a circle is enclosed inside the $2$-simplex. For $n=3$, solid angle ...
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1answer
73 views

Calculating a spread of $m$ vectors in an $n$-dimensional space

My question is regarding spreading $m$ vectors in an $n$ dimensional space such that the vectors are maximally distant from each other. For example, let us say I have a 2-D space, and 3 vectors, the ...
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1answer
70 views

Standing at the center of a cube and walking halfway to a wall - field of vision

In my python programming class one of the bonus problems is this: Suppose you are located at the exact center of a cube. If you could look all around you in every direction, each wall of the cube ...
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1answer
62 views

Do the $2^n$ hyper-octants of a $n$-sphere always have a $n$-dimensional right angle? Is $\pi/2$ only fundamental in $2$ and $3$ dimensions?

In $2$ dimensions, a $2$-sphere can be divided into $2^2 = 4$ congruent pieces, the $4$ quadrants, each of angle $\pi/2$ radians. In $3$ dimensions, a $3$-sphere can be divided into $2^3 = 8$ ...
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0answers
21 views

How are vertical angles always congruent?

Vertical angles are formed on opposite sides of two intersecting lines/line segments. I've also seen them as congruent angles because they have the same degree measurement, but how does this happen? ...
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2answers
36 views

Finding the angle of $\angle x$ and $ \angle y $

In the figure below, $ABCD$ is a rhombus and $ADE$ is a straight line. $\angle DAB = 51°$ and $\angle DCE = 42°$. what is the value of $x$ and $y$?
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1answer
2k views

How to find coordinates of 3rd vertex of a right angled triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
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2answers
912 views

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 ...
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1answer
57 views

Solid angle subtended in latitude-longitude maps

I need to scale a latitude-longitude map with the solid-angle each "pixel" subtend. How can I obtain the said solid angle starting from the $\phi$ and $\theta$ angles? Thank you very much
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2answers
469 views

Solving for Cos Exactly

How to solve $\cos(\dfrac{5\pi}{6})$ and $\cos(\dfrac{7\pi}{6})$ exactly? I couldn't use special triangles to solve this either.
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1answer
38 views

Is the Inverse of the Vectorised Solid Angle Equation for $n$ Circular Discs Continuous?

I have a continuous function$^{*1}$ that takes in 3 arguments, and returns 24 outputs. I want to know if the inverse of this function is continuous. The 3 input arguments are the x, y, and z position ...
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1answer
49 views

Generalisation of the concept of solid angles in curved space

I wonder if there are standard extensions of the concept of solid angle in curved space. There should be because physicists who study radiation in curved space would run into this kind of needs. If ...
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2answers
284 views

Calculating solid angles

For a project, I'm working on calculating solid angles using calculus, but when I test my formula for the solid angle of a regular tetrahedron, I end up with $0.4633$ when according to Wolfram, I ...
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0answers
341 views

Proof of solid angle theorem

I have a homework problem to prove about the solid angle. The book says: Let S be a smooth parametric surface and let P be a point such that each line that starts at P intersects S at most once. ...
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2answers
63 views

Calculate continous points on a circle circumference

Known details Circle radius r Origin (x,y) starting point in circumference say (u,v) I ...
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1answer
227 views

Needing Further Explanation of Knudsen's Cosine LAw

I'm reading over a paper by R. Feres and G. Yablonsky titled Knudsen's Cosine Law and Random Billiards, and I can't get around how they don't show directly how Knudsen's Cosine Law was derived. I'm ...
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1answer
247 views

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 ...
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1answer
553 views

how to derive relation between solid angle and surface area and the radius of sphere using definite integral?

how to derive relation between solid angle and surface area and the radius of sphere ? I know $s=r^2\Omega$ but how they got it using integral ?
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2answers
134 views

Instrinsic definition of concave and convex polyhedron

Is it possible to distinguish a concave polyhedron from a convex one by mesurements made only on its surface, without a reference to the 3d space around it?
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1answer
123 views

Transport theory basics: can't understand solid angles

I don't understand something in transport theory: $$P(x,\vec{w})=p(x,\vec{w}) \cos\theta \, dw \, dA$$ This is the number of particles flowing across a differential surface element in the direction ...
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1answer
518 views

Calculate angle of triangle

I need to calculate the angle between two sides, I have the length of A & B sides, but don't know how to find the angle... Both sides are the same length. I can get the start and end vectors of ...
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1answer
348 views

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 ...
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1answer
87 views

Solid angle definition - can it be seen shown using an image?

How do we show on a picture that a solid angle equals to this equation i found on a Wikipedia: \begin{align} \Omega =\!\!\!\int\limits_{\vartheta ...
2
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1answer
583 views

Angle between different rays (3d line segments) and computing their angular relationships

I have several positions (say A,B,C,..) and I know their coordinates (3d). From each point, if a certain ray is passing in a way to converge them at a given (known) point (say O). This point O ...
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3answers
846 views

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 ...
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0answers
183 views

Calculating the Solid Angle

$\textbf{Problem:}$ Consider we have a class $C^1$ parameterization $\psi:[a_1,b_1]\times[a_2,b_2]\rightarrow\mathbb{R}^3-\{0\}$ for the surface $S$. Also, consider that $S$ is such that the map ...
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1answer
38 views

Finding the angle.

first question here. I ran into a problem where my math skills are just not enough, probably it's simple, but I don't know how to approach it. I'll show you this graphic so you can understand what I ...
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1answer
136 views

Given an angle at a point A and given another segment BC, construct a point D so that the angle DBC equals the given angle at A.

I'm not sure if I understand this question right. Q. Given an angle at a point A and given another segment BC, construct a point D so that the angle DBC equals the given angle at A. Figure - ...
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2answers
248 views

Explain solid angle $\Omega=\int\int_S \frac{\bar r \cdot \hat n dS}{r^3}=\int\int_S \sin(\theta) d\theta d\varphi$

I am trying to understand this formula from Wikipedia here. It is a generalization of radian. I am trying to do unit check but I cannot see how the units match. Steradion must be dimensioless unit ...
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3answers
193 views

Relationship between angles in tetrahedron

Let's say I have a tetrahedron like this in image: Are the angles $\angle CAD$ and $\angle CBD$ equal in a general tetrahedron?
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4answers
4k views

Why is 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 $ d\theta d\phi $. No. It's $\sin\theta d\theta d\phi$, where $\theta$ is the inclination angle and $\phi$ is the ...