Questions tagged [geometry]

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

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How to comprehend the fact that parity is an improper rotation in the odd dimension, but not in the even dimension, physically?

Some "clarification" To begin with, I'm not even talking about relativity so, in the following, rotations always act on the Euclidean space or only the space subpart of the Minkowski space. ...
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Find location of geo satellite [closed]

I have some satellite antenna which is installed with offset. In order to verify the offset calibration to the right direction on first installation, I'd like to instruct to the antenna the satellite ...
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-2 votes
1 answer
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Prove dot product between two timelike vectors in Minkowski spacetimw [closed]

In the Minkowski 4-dimensional space-time $(\mathbb{M}^4,\eta)$ the dot product is: $a\cdot b = -a^0b^0 + a^1b^1 + a^2b^2 + a^3b^3 ~\qquad~ a = (a^0,a^1,a^2,a^3) ~~,~~ b = (b^0,b^1,b^2,b^3)$ Now ...
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Force to Inflate a ball underwater [closed]

How much force is required to fully inflate (with air) a beach ball that is 6 feet in diameter at depths of 200 feet underwater?
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3 answers
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Are there physics theorems that can prove maths theorems? Eg Pythagoras' Theorem

There's this recent post on maths overflow Which theorems have Pythagoras' Theorem as a special case? that has an answer by dxiv that appears to use a physics theorem to a prove a maths theorem, ...
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Please explain statement in a book on Loop Quantum Gravity

In a book by Carlo Rovelli on Covariant Loop Quantum Gravity, I struggle to understand a statement on Tetrahedron as follows: What is the dimension of the matrix? How to derive the given matrix ...
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2 answers
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On a minima problem in optics

I have trodding through a calculus textbook, more specifically — through a chapter on the methods of obtaining the extrema of functions using derivatives, including certain problems in optics (Fermat’...
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4 votes
2 answers
262 views

Why doesn’t horizon distance move exactly proportional to the height of the observer?

For instance if someone is 8 inches above the surface of the Earth, they can see approximately 1 mile to the horizon. However, if someone is viewing the horizon at an eye level of 5’5 they can only ...
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1 answer
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Why is the part of a sphere's area directly proportional to the square of its radius?

Solid angle in the book is explained in this way: "...Let $dA$ be a small area element of the surface of the sphere. If the points situated on the boundary of this area be joined to $O$ (the ...
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4 votes
3 answers
123 views

Why is the flatness problem called the "flatness" problem? What is its connection to geometry?

My understanding of the flatness problem is that it says that if we leave out dark energy and inflation, then the density parameter $\Omega(t)$ tends to $\infty$ or $0$ unless we have $\Omega(t) = 1$ ...
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-1 votes
2 answers
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How long of a wire do i need to construct a solenoid of $N$ turns? [closed]

Consider an MRI scanner with a hole diameter of $60 ~ cm$ and producing a field of $B_{0} = 1.5 ~ T$ and has a length of $50 ~ cm$. Furthermore, let us assume that the remanence of a permanent magnet ...
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How can we define length unambiguously when measurement of length is dependent on perspective?

Suppose I have two unmarked rulers, both of same length in one perspective. Let me keep them at some place in the room , and I view them from another. In the other point of viewing, the perceived ...
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How to calculate Distance $D$ to object from a camera with known $H$?

I am using algorithm to draw a bounding box around object when they appear in camera as shown in attached image. The main confusion is that i consider the object closer to camera when the Distance $D$ ...
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2 votes
1 answer
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Geometry: path length in atmosphere ("round" Earth)

I'm having trouble obtaining this formula. I'll paste the text from the book: Considering the curvature of the Earth ($R$ is the Earth radius) and a non-vertical direction (zenith angle $θ$), the ...
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Deviation of centre of gravity when adding and subtracting shapes

Is the distance that the centre of gravity moves when adding or subtracting the same shape from another shape the same? My teacher told me to assume that circle is being added instead of subtracted ...
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Finding the uncertainty in angles that need to be changed to radians, when dealing with trigonometry ratios?

I just wanted to ask about how we are supposed to do uncertainties in our experiment since in they y-axis is acceleration/cosx and on the x axis is tanx. Are we supposed to place error bars on both ...
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1 answer
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Is kinetic energy formula tightly related to Pythagoras theorem?

Consider 2 balls of mass $M$ traveling on the plain at speed $V$. One ball goes up and the other goes right. Let's associate them with the vectors $(V, 0)$ and $(0, V)$ to express their velocity and ...
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How did Al-Biruni know the height of a hill?

I knew of Eratosthenes famous experiment for determining the size of the Earth, but a recent question led me to read about another method by Al-Biruni. Like Eratosthenes' method, it required two ...
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9 votes
2 answers
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How did they know the radius of the earth in ancient times?

We've all heard how in ancient times they determined that the earth was round, and were able to calculate its radius based on the measured shadows of two separate sticks in the ground at noon. I am ...
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Geometrical Rotation and Rotation operator relation and ambiguity

In Cohen-Tannoudji's it is said (vol.1 page 694): $|\psi \rangle$ is the state of the system before the rotation. $|\psi ' \rangle$ is the state of the system after the rotation. $|\psi ' \rangle= R |\...
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Circles to squares

I always get lost in my thoughts on the mechanical physics of how things work or how it would work. Well I have my coffee cup here full of coffee, I flicked it and watched the waves close into the ...
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How to find a bend curve in 2D?

I am looking for a solution on how to find the (approxmiate) shape when bending a rigid-flex circuit board. Please see the abstract sketch below. I have two solid objects ($A$, $B$) which are ...
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Proofs for off-axis focal lengths of a concave mirror

I am reading "Fundamentals of Optics" the 4th edition, by Francis Jenkins. The book mentions a conclusion for off-axis concave mirrors: That is, depending on: mirror radius of curvature $r$...
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Energy density per solid angle?

How does this make sense? Energy density is usually a function of position (maybe time,frequency also) How could it also be a function of $\Omega$ , i.e the solid angle ? For context, $ u(\Omega)$ ...
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Analytic solution to geometric swinging problem

I came across this problem in the context of developing a game about a brachiating gibbon, swinging from handhold to handhold. (for context.) I was looking for a way to calculate what the gibbon's ...
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Distance between two points on Earth (arc length) doesn't increase with altitude (radius) according to GPS

Consider the following example: Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X. Point C has ...
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0 votes
4 answers
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Will it be $\vec{AB}+\vec{AC}=\vec{AD}$ or $\vec{AB}-\vec{AC}=\vec{AD}$? [closed]

The resultant of $\vec{AB}$ and $\vec{AC}$ is $\vec{AD}$. Now, which of the following is correct? $$\vec{AB}+\vec{AC}=\vec{AD}\tag{1}$$ $$\vec{AB}-\vec{AC}=\vec{AD}\tag{2}$$ I think $(1)$ is correct.
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2 answers
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Electric charge in compact space

Why in a compact space in the presence of an electric charge there must be the same charge with the opposite sign?
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Vector subtraction coordinates in rotated axes

I'm going through Anthony Zee's Einstein Gravity in a Nutshell and got confused on one of his points that seems to be a more general confusion about rotating axes. Zee first defines the rotation of ...
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2 votes
3 answers
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What is the difference between direction and angle?

For example, "angle the man must steer his boat" vs. "direction the man must steer his boat". Are direction and angle the same in this case? Or does it mean I have to find the ...
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Having trouble understanding Re-derivation of Young’s Equation

I was going over a section 2.1 of this article (regarding Young's Equation and for some reason I wasn't able to derive E10 from E9 (see image below). I know it is more about the trigonometry of the ...
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0 votes
1 answer
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Lateral Force formula in sail boats

This is my first time asking a question here so I'm sorry if it's not supposed to be here. I am working on a project relating to the forces on a small dinghy I own. More specifically, I'm looking into ...
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0 votes
1 answer
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Will the equations derived for dark and bright fringes remain the same in this case too, or will there be changes?

We know in YDSE experiment we calculate position of white and dark fringe by calculating the path difference using Pythagoras theorem and equating it to $n\lambda$ and $\frac{2n-1}{2}\lambda$ ...
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Question about volume of water [closed]

As shown in the figure, the lead straight cross-section of a reservoir and two mountains is shown. The rectangular groove at the bottom represents the reservoir, and the isosceles triangle on both ...
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0 votes
2 answers
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How is the formula $R=l^2/6h+h/2$ for calculation radius of curvature of a spherical surface is derived? [closed]

The problem with the derivation is that in one figure the point M is taken the bisector of AB while in the second one it is completely out of the line AB. So for AM.MB=OM.MZ to be true the chords AB ...
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0 votes
1 answer
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Where would the centre of gravity be for 4D shapes?

Say, hypothetically, we took a hypersphere where would the centre of gravity be for such an object.
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2 votes
0 answers
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Geometric Entropy & the Partition Function for Conical Deficits

In their paper, Callan and Wilczek claim to derive from the thermal entropy $$S_\text{thermal} = -\left(\beta\frac{\partial}{\partial\beta}-1\right)\ln(\mathcal{Z})$$ a geometric entropy which is ...
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What is "antipodenpunkte" (this is German)? [closed]

What does "antipodenpunkte" mean? If you can't find a word about Einstein on the Internet, who should know it? "In addition, the question arises: can we see stars very close to our ...
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1 vote
1 answer
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Why Laplace-Runge-Lenz vector in a circular motion is $0$?

This vector is the sum of two vectors, and I understand why their direction is opposite, but I don't understand why their magnitude is the same. I know that the direction of this vector is always the ...
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How many pixels could an image of Proxima b taken by James Webb have?

I know it's very difficult for the James Webb to image Proxima b without a coronograph (I have been told by NASA scientists that they don't know yet whether they will be able to do so), but I wonder ...
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1 answer
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What is the shape made by a half-filled ball?

An inflatable ball is made of a thin, inelastic, but bendable material. When fully inflated, it has a radius $R$. If the ball is filled half full with air, and is then continuously pushed underwater ...
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4 votes
1 answer
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The metric of world lines in Newtonian and Galilean spacetimes

Consider a flat Newtonian or a flat Galilean 2+1 spacetime. So, mainly a flat 2D Euclidean space, evolving over time, where each time-slice is connected with the next one by a world line. Like in this ...
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1 answer
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Centre of mass of a quarter of disc (doubt in my solution) [closed]

Can please anyone tell where am i wrong in this approach of solving this question.. The ans i am getting is (4 root2)/pi but ans is (4 root2)/3pi
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0 votes
1 answer
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Inverse haversine formula - Change in geocentric coordinates from great circle distance and bearing?

I think this question best fits here - it's reasonably mathematical in nature, but is formulated as a practical kinematics problem; I'm happy to move it if I'm wrong. I'm trying to build a plane ...
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2 votes
4 answers
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Why do we need position vectors?

Position vector is a vector which defines location. It is also the only kind of vector which can't be moved. But coordinates can also do the same work. They also define position. So why do we need ...
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-1 votes
1 answer
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How to find out whether more than 2 vectors are coplanar? [closed]

Can you please explain me how to find out whether 3 or more vectors in $a\hat i + b\hat j + c\hat k$ format? I know that for 3 vectors, their dot product should be 0 but what is the reason behind this?...
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2 votes
2 answers
168 views

Is it possible to see the back of an object traveling at relativistic speeds?

A while ago someone told me it's possible to see the opposite side of an object when it's moving at relativistic speeds. I don't remember where I heard this, and they didn't give me an explanation, ...
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2 votes
1 answer
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How can two beams in the Michelson interferometer come back at the same time?

I am trying to understand this concept from my textbook. According to the figure below, the time difference between the two paths $\Delta t=0$. But why is this the case? Looking at the picture, the ...
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0 votes
1 answer
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How would a rainbow from a divergent light source look like?

Many explanations of the shape of rainbows as constant angle reflections do assume the light, incident on rain droplets, to be a parallel beam of light rays. That might for the sun-earth system be ...
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Lambert's cosine law in 3D, why $\cos\theta$ instead of $\cos^2\theta$?

I'm reading the book An Invitation to 3-D Vision and I don't understand a part of the explanation regarding the "basic radiometric image formation model". Looking at the Wikipedia page of ...
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