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

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0answers
28 views

Space formed by dot products of three vectors [migrated]

Suppose I have 3 3D unit vectors $\mathbf{v}_1$, $\mathbf{v}_2$, and $\mathbf{V}$. I define the three corresponding scalars $u_1=\mathbf{v}_1 \cdot \mathbf{V}$, $u_2=\mathbf{v}_2 \cdot \mathbf{V}$, ...
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1answer
36 views

How to calculate the horizon line of a satellite?

I need an equation to calculate a list of Earth-centered, Earth-fixed (ECEF) XYZ coordinates on the earth that represent the visibility limit of satellite given its ECEF XYZ coordinates. For any ...
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4answers
219 views

Why drops form spheres?

Consider a drop of water floating in an inertial frame in STP air (e.g., the ISS). Intuitively, the equilibrium shape of the drop is a sphere. How would one prove that? Is it equivalent to showing ...
5
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1answer
79 views

Gravity on and inside a planet-sized bi-lobed body

Admittedly, for yet another science-fiction project Say I have a planet-like body shaped like a sphere with a torus subtracted out of it, leaving a sort of "apple core" shape. Firstly, is the ...
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0answers
14 views

Reference Request: Resources on Affine Geometry [on hold]

I am reading Gravitation by Misner, Thorne, and Wheeler and they mention the term affine geometry. Like for many math topics, I was hoping there was just a go-to textbook on the subject but I haven't ...
4
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2answers
196 views

Can Minkowski spacetime be redefined as a non-flat riemannian manifold?

Minkowski space time is defined in terms of a flat pseudo-Riemannian manifold. I have wondered if it can be redefined as Riamannian manifold and in the case what type of curvature would there appear. ...
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1answer
105 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
400 views

Tensor of inertia

The tensor of inertia of a solid sphere is $I_{ii}=\frac{2}{5}MR^2$ about an axis passing through its CM. Why would the tensor of inertia of each hemisphere about that axis be ...
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2answers
1k views

Is it possible to calculate the length of a tube if the 'near' diameter and 'far' diameter are known?

When looking down and through a length of tube, the 'far' diameter is less than than that diameter that is closest to the viewer's eye. If, for example, the tube's internal diameter is measured at ...
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0answers
37 views

Maps on manifolds

I am reading the SUPERGRAVITY textbook of Freedman and Van Proyen. I am reading that in the sphere we can introduce two patches that their union covers the whole sphere. Ok, I understand why we need ...
2
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2answers
322 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 ...
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0answers
139 views

Time Dilation Geometry

I was recently exploring time dilation from Gravity and from velocity and I came up with an interesting derivation that I have not seen before. I was wondering if there is a paper published showing ...
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2answers
61 views

How many are the points which are $n$th nearest to a certain point in a hexagonal lattice

Suppose there are infinite points arranged as hexagonal lattice. The question is the one as the title. First we choose a point called $A$. Then when we count the $n$th nearest points to $A$, what ...
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2answers
133 views

Where to take $\theta$ (theta) in a triangle?

This is an inclined surface having and unit vector A. The perpendicular and horizontal surfaces are the components of the inclined surface. We want to find out that how much flux will pass through ...
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0answers
22 views

Gaze tracking with the pupil-glint response how does the geometry work?

I am struggling to understand the geometry behind the pupil glint response (see above) (With only one light source). I understand that you use the glint to find the corneal centre (Centre of corneal ...
0
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1answer
51 views

Do lower dimensional objects actually exist? [duplicate]

Take for instance a arbitrarily chosen 2-dimensional rectangle, having a length of $x$, a width of $y$ and height of 0. (Or take for instance a 1-dimensional line. Or take for instance a 0-dimensional ...
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0answers
20 views

Calculating errors?

Hollow cylinder of length l, inner and outer radii r1= 2.5+-0.3mm, r2= 5.5+-0.3mm, density= 7.88g/cm^(3), mass density= p, mass= (pi)xpxlx(r2^(2)-r1^(2)). What is the mass?
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0answers
109 views

What is the fourth dimension of a Tesseract? [migrated]

Is the fourth dimension of the Tesseract time? That is why it is represented as a moving 3D structure on Wikipedia? I am asking because I have trouble understanding what it is.
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2answers
37 views

How would the representation of a circle in 4 dimensions? [closed]

We know that the representation of a square in three dimensions is a cube, and in four dimensions is a tessaract or, hyper cube. With this, How would be the representation of a circle in four ...
0
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1answer
30 views

In layman's terms, why would frame dragging affect precession of nearby object?

My question is really about the gravitomagnetic frame-dragging and the Lense Thirring effect. My question is not whether the frame dragging effect exists but rather it's manifestation in affecting ...
5
votes
4answers
659 views

How did “no prior geometry” father 50 years of confusion?

I've come across this quote attributed to Misner, Thorne & Wheeler from their book, Gravitation: Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior ...
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2answers
451 views

Geometrical interpretation of complex eigenvectors in a system of differential equations

Let's consider a system of differential equations in the form $$\dot{X} = M X$$ in two dimensions ($X = (x(t), y(t))$). In the case that $M$ has real values, it is easy to give a geometric ...
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1answer
59 views

What solution do I need to use in this question? [closed]

Question: Jojo and Albert left the same point and hike in different directions. their displacements are $\Delta x= 4$ km and $\Delta y= 6$ km. How far are they from each other?
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1answer
62 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, ...
0
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1answer
39 views

How corner atom of a cube is shared by 8 atoms? [closed]

Since each atom is at the corner of a cube and 8 cubes meet at each corner. Therefore each atom is shared by 8 cubes. But I don't visualize it, so please tell how I Visualize it.
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1answer
29 views

How do you compute for the center of a rotation of a bicycle?

I know how to find the center of a rotation, like where you connect the side and make bisector with the help of your compass. But in a situation like this how do I get the $a_2$? I'm doing an advance ...
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1answer
29 views

$y$-component of center of a cube that rolls without slipping [closed]

For the figure above, let $O$ be the origin point for $x,y$-axis, and $+x$ goes to the horizontal right direction while $+y$ direction goes to the up direction. The figure has one circle and one ...
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2answers
107 views

What is the difference between the shapes of molecules with different isotopes

I'll explain my question on example of water molecule. Let us have three water molecules: normal water $H_2 O$, heavy water $D_2 O$ and semiheavy water $HDO$. Is there any difference between the ...
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0answers
33 views

Increasing the volume of a object in a photograph

I have two images which look like this --- These images are two views of a calibration object --- the white die that you see hanging in the middle. The problem is that, the die is too small and ...
3
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2answers
76 views

Kepler's First Law of Planetary Motion: Whats at the other focus? [duplicate]

Kepler's first law states that the orbit of every planet is an ellipse with the Sun at one of the two foci. Is there anything significant about the location, or any unusual properties about the other ...
1
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1answer
70 views

Can one calculate the distance from Earth to the camera, given a single picture?

Given an image of (part of) a sphere of known size, can one calculate the distance to the camera based on the relative size of the sphere in the image? My instinct is "no" unless you have some fixed ...
4
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2answers
184 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 ...
5
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0answers
32 views

Motivating Irreducibility of Hilbert Space for Quantization Axioms

In the context of geometric quantization, we usually look for a map from the Poisson algebra of classical observables to the algebra of quantum observables (or rather, a sub-algebra of the classical ...
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3answers
86 views

Finding the Direction when resolving a Vector

When you are finding the resultant of a vector I understand that to find the magnitude you use Pythagorean Theorem and I understand that to find the direction you are going to use $\tan^{-1}$ (or ...
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1answer
3k views

Minimum height of mirror required to view image

I wanted to know the minimum height of mirror required to be able to view a complete image of a person. I considered the following setup: $HF$ is the person in question. $H$ denotes the head, $F$ ...
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3answers
41 views

Deriving relationship between distance and angle for field of a line charge

The relationship $r\,\mathrm{d}\alpha = \mathrm{d}x \cos(\alpha)$ is not obvious to me. In fact, when I look at it, I think it should be $r\,\mathrm{d}\alpha \cos(\alpha) = \mathrm{d}x$. Can someone ...
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1answer
34 views

Centre of mass as a function of distance [closed]

I have searched it on Google and also visited the HyperPhysics website to find out but all they seem to offer world centre of mass of continuous and uniformly bodies.My school textbooks are of no help ...
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5answers
44 views

Particles passing through a surface

Today is the day I ask silly questions : The book says the particles passing through the surface $dS$ are the ones contained in the cylinder of volume $dS.v.dt.cos(\theta)$ but I really don't see ...
9
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1answer
224 views

Cutting a circle and moving endpoints

A metal (or otherwise, suitably elastic) circle is cut and the points are slid up and down a vertical axis as shown: How would one describe the resultant curves mathematically?
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2answers
100 views

Why does the $L_2$ norm give the shortest path between 2 points?

Why not the $L_1$ or $L_3$ distances? Is there some deep reason why the universe (at least at human scales) looks pretty much Euclidean? Could we imagine a different universe where a different ...
1
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2answers
111 views

Number of revolutions of a rolling coin [closed]

Take two quarters and lay them on a table. Press down on one quarter so it cannot move. Then, starting at the 12:00 position, roll the other quarter along the edge of the stationary quarter. How many ...
0
votes
1answer
39 views

Error in measuring distance ignoring curvature of Earth [closed]

Suppose you model distance as a flat 2d plane rather than a curved surface. Given that the radius of the Earth is about 6400 km, approximately how far must you travel before the relative error ...
0
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2answers
71 views

How does the human eye knows how far the object from which the photon was reflected?

A photon is emitted from a source and reflected off an object (or objects) until it hits the human eye. The color of the object we see depends on the photon wavelength. If photon travels with constant ...
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6answers
1k views

Do perfect spheres exist in nature?

Often in physics, Objects are approximated as spherical. However do any perfectly spherical objects actually exist in nature?
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0answers
44 views

Moment of Inertia of Polygons in the Plane [closed]

I was reading this link, which describes a method of finding the moment of inertia of a general convex polygon by splitting it into triangles. I then realized I have no idea on how to derive a such a ...
0
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1answer
37 views

How do I find the moment of inertia of a regular $n$-gon? [closed]

Of a regular $n$-gon with radius $R$ and mass $M$. Any hint to solving would also be acceptable. The result I'm looking for is $$I_{CM} = (1/2) MR^2 (1 - (2/3) \sin^2(\pi/n)).$$
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1answer
164 views

$c/a$ ratio for an ideal hexagonal close-packed (HCP) structure [closed]

Show that the $c/a$ ratio for an ideal hexagonal close-packed (HCP) structure is $\left(\frac{8}{3}\right)^\frac{1}{2} = 1.633$. I believe $a$ is the length of $a_1$ and $a_2$. I figured that ...
1
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1answer
68 views

Path of wheels of a bicycle

Why are the wheels of a bicycle moving in concentric circles with the center O? I know that the velocity of the back wheel is parallel to the frame of the bicycle and the velocity of the front wheel ...
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0answers
47 views

Bending of a pipe filled with sand

There is this industrial pipe made out of steel that is 1 meter in diameter and 6 meters long (and volume 4,71 cubic meters). The thickness of the pipe "wall" is 4 centimeter. The pipe is filled ...
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0answers
14 views

What are some structures which minimize surface stress upon expansion

Intro I am currently researching into a problem involving the cyclical expansion and contraction of a 3 layered structure comprised of an internal metal encased by another metal encased in an outer ...