To be used for questions on geometry closely pertaining to physics. Includes differential geometry and euclidean geometry.

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1answer
81 views

find distance using a light line

Let's suppose we have a particular light frequency emitter and relative sensor array, and that there is no external source of this light. This emitter has a know angle respect the receiver, and emit ...
2
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2answers
255 views

Finding the illuminance from a triangular light source

Since most light sources in games are point-like, it's pretty difficult to approximate area light sources with point sources. As triangles are a universal form to represent 3D models (thus area light ...
2
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1answer
107 views

Why is the inertia ellipsoid of a higher symmetry than the rigid body?

I was always puzzled by this fact. A uniform cube has a sphere-shaped inertia ellipsoid. The sphere has a higher symmetry then the cube. Is there any deep reason or implication behind it?
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2answers
71 views

A simple way of calculating Euler Angles from Rotation Matrix — help!

This is a follow up of this question : I have the rotation matrix $$ \left( \begin{matrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & ...
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0answers
32 views

Definition of a geodesic ball? [migrated]

I think it goes along the lines of: a ball made of a series of flat sides. Also is a geodesic ball and geodesic dome the same thing?
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1answer
85 views

Inertia tensor of a spherical cap

I'm trying to calculate the inertia tensor of a spherical cap (a piece of a sphere) like the one shown below. The origin (not shown) is located at the center of the whole sphere and the axes ...
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2answers
36 views

Euler angles derivation

I have been trying to grasp the idea of Euler angles for a while. Can anyone point out if my understanding is correct or not. Situation: We have 3 axes known as principal axes of inertia which define ...
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1answer
32 views

Leg Press & Actual Lifted Weight [duplicate]

I was doing leg press at the gym today and was curious how much weight I actually lift when I do the exercise as compared to when I do a squat. Suppose I load $w_L$ onto the machine, which has an ...
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1answer
40 views

Geometric Interpretation of Rotated basis of Hamiltonian and collective Dicke states

Suppose I start with a basis of states for a two spin-1/2 particle system, namely, $\{\left|\uparrow\uparrow\right\rangle, \left|\downarrow\downarrow\right\rangle, ...
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1answer
86 views

Structure factor of crystals (X-ray crystallography)

How can one prove that the degree of each node in a distance graph must be at least four in order to obtain a unique solution to an exact distance geometry problem with sparse distance data? The ...
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1answer
72 views

Relative strength of a dome

Is there a rough way to determine how the height of a dome affects the load that dome could support? For instance, assuming the bases of two domes are 24" in diameter, and one dome is 2" high while ...
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1answer
235 views

How to calculate roll, yaw and pitch angles from 3D co-ordinates (Euler Angles)

I have digitized a video of a flying fly in a 3-dimensional space. At all instants I know the x, y, and z co-oridinates of the following points on the fly's body --- The points are my choice, and ...
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1answer
18 views

random particles on a grid: Effect of increasing density on distance between them

Say I have two boxes which both contain, say, 25 red particles (as shown in picture). These particles are randomly placed in a 2d grid, and in one the total area $A_{1}=20000$ and the other has area ...
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3answers
4k views

Why can't a piece of paper (of non-zero thickness) be folded more than $N$ times?

Updated: In order to fold anything in half, it must be $\pi$ times longer than its thickness, and that depending on how something is folded, the amount its length decreases with each fold differs. ...
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1answer
56 views

Measuring the nearest order of magnitude

The world's largest ball of a string is about $R=2 m$ in radius. To find the nearest order of magnitude, what is the total length $L$ of the string in the ball? I have tried this in the following ...
3
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1answer
143 views

Can a very small portion of an ellipse be a parabola?

We consider that when a body is projected from any height from the earth surface with a speed lesser than the orbital speed ( tangentially to the earth surface at that point.) it follows an elliptical ...
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1answer
140 views

Characteristic length of a triangle [closed]

I am reading a paper on collision detection in cloth simulation, please help me understanding following lines written in the paper : To check if a point x4 is closer than some thickness h to a ...
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4answers
181 views

Electric Field due to a disk of charge. (Problem in derivation)

This might be a really silly question, but I don't understand it. In finding the electric field due to a thin disk of charge, we use the known result of the field due to a ring of charge and then ...
2
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2answers
6k views

Determining the center of mass of a cone

I'm having some trouble with a simple classical mechanics problem, where I need to calculate the center of mass of a cone whose base radius is $a$ and height $h$..! I know the required equation. But, ...
5
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1answer
202 views

Why is $S^1\times\mathbb{R}^{n-1}$ the topology of $AdS_n$?

Anti-de Sitter $AdS_n$ may be defined by the quadric $$-(x^0)^2-(x^1)^2+\vec{x}^2=-\alpha^2\tag{1}$$ embedded in ${\mathbb{R}^{2,n-1}}$, where I write ${\vec{x}^2}$ as the squared norm ${|\vec{x}|^2}$ ...
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1answer
74 views

Geometry in diagonal matrix and inertia tensor

For this problem, can anyone explain to me why when $x_1$ axis is aligned with the diagonal of the cube, the resulting inertia tensor will become diagonal? How to interpret this result geometrically? ...
0
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1answer
93 views

Moment of Inertia of a sector of a circle [closed]

I am trying to find the moment of intia about its centre of a sector of a circle of radius $a$, mass $m$ and angle $\pi/3$. I have found the answer it is $\frac{1}{2}ma^2$ but originally tried a ...
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0answers
66 views

Transforming components of the angular momentum operator

Let me introduce the problem: In a two electron fixed nucleus problem the "body" is the atom, whose electrons are located relative to the nucleus by the coordinates $r_1$ and $r_2$, and the angle ...
3
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2answers
221 views

Differentials in Spherical Shell - Maxwell Distribution

In explaining the Maxwell distribution of molecular speeds, my pchem textbook uses the following figure: We are basically trying to find the probability of having a particle with a speed $u$ between ...
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3answers
63 views

How can I compute the angular velocity of a triangle formed by three particles knowing their instantaneous positions and velocities? [closed]

I have a set of trajectories of three particles and their instantaneous velocities. I would like to compute the 3 components of the angular velocity pseudovector of the fictive triangle formed by ...
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6answers
2k views

Experimental evidence of a fourth spatial dimension?

As human beings, we observe the world in which we live in three dimensions. However, it is certainly theoretically possible that more dimensions exist. Is there any direct or indirect evidence ...
5
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1answer
143 views

What's a geometric explanation for exponential-falloff fundamental forces?

Gravity and electromagnetism are inverse-square laws. This makes geometric sense -- if you build a spherical shell around a lamp then a shell with twice the radius has four times the surface area and ...
5
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0answers
108 views

Why do we need non-trivial fibrations?

I am currently reading this paper. I understand how the Bloch sphere $S^2$ is presented as a geometric representation of the observables of a two-state system: $$ \alpha |0\rangle + \beta |1\rangle ...
10
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1answer
410 views

Can masses move in 2+1 gravity?

I would like to understand basic concepts of the general relativity in 2+1 spacetime. As far as I know, GR predicts that such a spacetime is flat everywhere except for the point masses which create ...
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2answers
87 views

How much of Minkowski spacetime structure can be recovered from its causal structure?

A beginner's question: I have always understood that (four-dimensional) Minkowski spacetime can be recovered up to a constant factor—i.e. 'up to a dilation' or 'up to global scale'—from its causal ...
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2answers
116 views

What is the notion of a spatial angle in general relativity?

Is there a notion of spatial angles in general relativity? Example: The world line of a photon is given by $x^{\mu}(\lambda)$. Suppose it flies into my lab where I have a mirror. I align the mirror ...
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0answers
49 views

Rigid rectangle in Schwarzschild

Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial ...
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1answer
77 views

How to calculate spatial distance in space-time?

Pinning two test particles at two different points in space, how can I calculate their spatial distance, when the geometry is given by the Schwarzschild metric? Let's say particle 1 is pinned at ...
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2answers
173 views

Is the triangle stronger?

I have a friend who cedes that a framework that is a triangle would be stronger than a square framework. But he maintains that a square solid would not be stronger than a triangle solid. Ie a set of ...
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2answers
120 views

Are signal fronts in a beam not at rest to each other?

I'd like to investigate how the notion of "mutual rest" might be applied consistently, but distinctively, in the following thought experiment: Consider a light source ("$A$") which directs a beam ...
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2answers
3k views

The resultant of two forces acting at any angle?

I am studying about forces as vectors. And they give me this equation: $c^2 = a^2 + b^2 - 2ab \cos C$ Can anybody explain me the second part of the equation? I perfectly understand $c^2 = a^2 + b^2$ ...
0
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1answer
70 views

In Minkowski space, why does the hyperboloid appear to each observer as a circle whose radius is increasing faster than the speed of light?

I read the assumption in the above question in the paper Hyperbolic geometry on a Hyperboloid by William F. Reynolds (see here, page 444), but it was not clarified further (the discussion was rather ...
3
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2answers
82 views

Scattering geometry question

While reading up on light scattering I came across this slide: My vector maths is a bit rusty and I am having trouble understanding the last term (scattering geometry). What is the significance of ...
3
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0answers
116 views

Geodesic distance in de Sitter space

Consider $N$ dimensional de Sitter space embedded in $N+1$ dimensional Minkowski space: $$\eta_{\mu\nu}X^\mu X^\nu=1, \hspace{1cm}\eta_{\mu\nu}=\text{diag}(-1,1,\dots,1)$$ where I set $H=1$ for ...
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1answer
73 views

Physics of Skiing: Ideal Carving Equation

I am conducting a research paper on the physics of skiing, specifically how ski parameters affect the ski's ideal carve. I have come across this paper (abstract link to arXiv paper), which is ...
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1answer
363 views

When to use Cosine or Sine? [closed]

Three blocks of masses $M_A=15.0 \; \text{kg}$, $M_B=12.0 \; \text{kg}$, and $M_C=8.0 \; \text{kg}$ sit next to each other on a frictionless surface. A force $F=54.0 \; \text{N}$ acts on Block $A$ ...
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0answers
13 views

Did a racer on a “restless” course race at some particular average velocity? (2: Starting gate and finish line being rigid to each other)

Consider two participants, a starting gate ($A$) and a finish line ($B$), in a (sufficiently) flat region, both undergoing uniformly accelerated motion, such that $A$ finds constant ping duration ...
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0answers
20 views

Did a racer on a “restless” course race at some particular average velocity? (1: Starting blocks, finish line moving uniformly wrt. each o.)

Consider two participants, $A$ and $B$ moving uniformly wrt. each other; first having approached each other, then having met and passed each other, and eventually having separated from each other such ...
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0answers
80 views

Electric Potential of a Cube Made of Point Charges

I'm trying to find the potential energy of multiple geometric shapes made entirely out of point charges. This particular shape is a cube made out of two different point charges, A and B, each ...
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2answers
146 views

How far would I have to go to see a fully rounded Earth?

How far would I have to go to see a fully rounded Earth? Recently, I saw a video on Youtube in which a sky diver called Felix Baumgartner ascends to $120,000$ feet (= $39$ miles) in a stratospheric ...
2
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0answers
67 views

If two ends were a certain “length” apart were they therefore at rest (or at least rigid) to each other? [closed]

Considering the definition of the SI unit of "length" [1] and [2 (" method a.")] I'm missing any requirements about the two "ends" of the required "path travelled by light" being "at rest to each ...
2
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0answers
38 views

Can a value of “length, in meters” be attributed to a pair of ends which are rigid (but not at rest) to each other? [duplicate]

The definition of the SI base unit "metre" [1] doesn't seem to rule out explicitly that a certain value of "length, in meters" could be attributed to a pair of ends which are rigid to each other, but ...
15
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2answers
374 views

Generalized Complex Geometry and Theoretical Physics

I have been wondering about some of the different uses of Generalized Complex Geometry (GCG) in Physics. Without going into mathematical detail (see Gualtieri's thesis for reference), a Generalized ...
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3answers
378 views

What's the dimensionality of a solid angle?

I haven't seen this explained clearly anywhere. Solid angles are described usually as a fraction of the surface area of a unit sphere, similar to how angles are the fraction of the circumference of a ...
4
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3answers
302 views

Calculating the Center of Mass

We have a homogeneous body that looks like this: I have tried dividing the body into different parts using the following definition: R g * A = R 1 * A 1 + ... R n * A n I was thinking I could ...