Questions on spherical coordinates, a three-dimensional coordinate system where a point is represented in terms of its distance from the origin, and its latitude and longitude angles (or complements thereof).

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11 views

New Axis calculation

I have a task whereby I use an accelerometer to calculate acceleration for a vehicle. The problem I am attempting to solve is to allow the accelerometer to be in any oriertaion. Basically I have a ...
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37 views

Volume element in spherical coordinates

In spherical coordinates, we have $ x = r \sin \theta \cos \phi $; $ y = r \sin \theta \sin \phi $; and $z = r \cos \theta $; so that $dx = \sin \theta \cos \phi\, dr + r \cos \phi \cos \theta ...
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10 views

vector components and dot product with unit vector

$E_{0}\hat z=\vec E_{0}rcos\theta=E_{0}cos\theta\, \hat r$ and $E_{0}cos\theta\, \hat r\circ \hat r =E_{0}cos\theta$ This just doesn't look right to me for some reason...
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30 views

Showing the uniqueness of the solution to some problem?

I have faced problems proving all kinds of uniqueness theorems but this one I've come across seems particularly tricky to me. Can you help me? The function g(x,y,z) is zero for r²>a² (where ...
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1answer
29 views

Curl of a function with only angular dependence

Let a function in spherical coordinates $$\vec F(\vec r) = \int{ d^3\vec r\,' \vec j(\vec r\,') } \,e^{-ik\hat r \cdot \vec r \,'}$$ Where $\vec j$ is a vector function. So $\vec F$ only depends on ...
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1answer
20 views

Find a vector in cartesian coordinates given its relative location to another vector in spherical coordinates

Here is my problem: -I have an arbitrary normalized vector N in cartesian coordinates -I am trying to find normalized vector M, also in cartesian coordinates -I am given the azimuth and polar ...
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45 views

How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
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54 views

A basic question on surface area of spherical cap of sphere

Consider a sphere of radius 1. Now chop a spherical cap with latitude line $\phi$ at the bottom of the cap is removed from top (say $0<\phi<\frac{\pi}{2}$). I want to know the surface area of ...
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49 views

Mean value of a function over the n-sphere's surface.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
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206 views

Function psi, vector potential, satisfying conditions

Using spherical polar coordinates ($r, \theta, \phi$) verify that the vector $F = r^{-2}e_r$ is solenoidal. Find the function $\psi(r, \theta)$ such that $A = \frac{\psi(r, \theta)}{rsin ...
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201 views

Volume enclosed by two spheres (triple integral, cylindrical coordinates)

The question: Find the volume of the solid enclosed by the sphere $x^2 + y^2 + z^2 - 6z = 0$ , and the hemisphere $x^2 + y^2 + z^2 = 49 , z ≥ 0$ I set up the triple integral ...
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91 views

Conversion of covariance matrix from Cartesian to Spherical coordinates for integration

I have to perform a convolution of a function in polar coordinates $\rho(\textbf{x}) = \rho(r,\theta,\phi)$ with a function $P(\textbf{x}) = P(x,y,z)$ in cartesian coordinates. $\int ...
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19 views

Projecting spherical components of a variable point to the unit vector of a fixed point

Consider the following vector function in spherical coordinates: $\mathbf{v} (r, \theta, \phi) = V_{\phi} (r, \theta, \phi) \mathbf{a}_{\phi} = A \delta(r - k) \displaystyle \delta \left( \theta - ...
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2answers
125 views

Center of mass of one octant of a non-homogenous sphere

Find the center of mass of that part of the sphere $x^2+y^2+z^2 \le a^2$ having $x,y,z \ge 0$ (that is, the part in the first octant) With density given by $\rho(x,y,z)=(x^2+y^2+z^2)^{3/2}$ It ...
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57 views

Finding the coefficients of a partial differential equation after a change of coordinates.

I'm stuck in one of the mathematical steps of my physical problem. I've been following the derivation of my equations (starting at section 4) from this article Symmetric Euler-Angle Decomposition of ...
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91 views

Calculate Phi & Theta in Sphere

My target is to calculate phi and theta in a sphere. I built a panoramic viewer. In this viewer the user can load a picture. This picture will be load onto an sphere with a radius of 50. The user ...
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1answer
29 views

Sphere to Caresian Coordinates and vis versa

I want to convert from Cartesian to sphere coordinates and back Here is my code ...
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1answer
151 views

How to calculate this integral in 3 dimensions involving the Dirac delta function?

How would I go about calculating the integral $ \int d^3 \mathbf r {1\over 1+ \mathbf r \cdot \mathbf r} \delta(\mathbf r - \mathbf r_0) $ where $\mathbf r_0 = (2,-1,3)$ My attempt so far: I have ...
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50 views

How to rotate point on a spherical coordinate

I am looking for formula that shows hot to find the spherical coordinates of a point after the axis rotated. Assume that I am on earth and I have the coordinate of a point in lan/lot (which I believe ...
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61 views

A problem on vector calculus on spherical coordinate systems

I'm wondering whether I could get some idea on a problem I've been working on, here it is: h and g are two functions defined on the surface of a unit sphere, and their values depend on the ...
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31 views

New direction of propagation of light

\begin{align} \mu'_x & = \frac{\sin\theta(\mu_x \mu_z \cos\varphi - \mu_y \sin\varphi)}{\sqrt{1-\mu_z^2}}+ \mu_x \cos\theta \\ \mu'_y & = \frac{\sin\theta(\mu_y \mu_z \cos\varphi + \mu_x ...
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101 views

Find a common point that three lines meet.

I have a base 2D triangle with 3 lines coming out of each vertex with their own coordinate point (xyz) and a set distance, is it possible to calculate the specific point that they should meet? I also ...
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1answer
63 views

Finding a 3D co-ordinate using triangluation

I'm trying to find a real world coordinate where 3 spheres collide and interact. At the moment I have been able to set up my triangulation equations so that I can work out the 2D position of where my ...
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1answer
115 views

Interpolating geographic coordinates

I have two geographic coordinates. Let's call them $A$ and $B$: A = latitude 41.34759, longitude -75.77415 B = latitude 41.34769, longitude -75.77404 My unknown ...
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1answer
71 views

Why there is discontinuity at Zenith in Spherical-coordinate system?

I tried to plot the following function in spherical-coordinate system : $$ r(\phi,\theta)=\left(\frac{\sin\phi}{\phi}\frac{\sin\theta}{\theta}\right)^2$$ (definition/references for ...
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26 views

Cross product in spherical coordinates [duplicate]

Does anybody know how the cross product in spherical coordinates looks like? So I have a vector with $ae_r+be_{\theta} + ce_{\phi}$ with another vector $de_r+fe_{\theta} + ge_{\phi}$, how does there ...
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24 views

Optimization: How can I limit parameters variability?

I'm optimizing the normal of a 3D plane minimizing the re-projection error (with the difference of the intensity of some pixels in two images that corresponds of some points that lies on that plane). ...
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93 views

Transforming an integral over $R^n$ to a radius and a directional vector (aka Spherical-Radial)

In several papers by John Monahan and Alan Genz there's mention of a spherical-radial transformation: $$ \int_{\mathbb{R}^n} f(\mathbf{x}) ~\mathrm{d}\mathbf{x} = \int_0^\infty ...
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55 views

“Great Circle” distance [duplicate]

Given two points on a sphere, then the "great circle distance" between two points is the length of the smallest arc of a great circle containing both points. Assume that $\Sigma$ is a sphere of radius ...
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82 views

Nodes in spherical equations and graph matching

The graph for this spherical equation is, (equation no 44) $$\frac{d^2 S}{d \rho^2}+ \frac{D-1}{\rho}\frac{dS}{d \rho}-S +S^3=0.$$ What I didn't understand here is $S_0, S_1,S_2$ and $S_3$. ...
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59 views

Addition of spherical surface vectors

I'm making a planet simulator, which makes much use of a sphere. I'm trying to figure out how to represent and manipulate vectors on the surface of the sphere. Currently, my coordinates are all ...
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1answer
95 views

Law sines in Spherical Triangle $\rightarrow$ Law sines in plane triangle

Could any one tell me how to estimate or get law of sines in Spherical Triangle to The Law of Sines in Plane Triangle? i.e $\frac{\sin a}{\sin A}=\frac{sin b}{\sin B}=\frac{\sin c}{\sin C}$ to ...
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103 views

two points on a unit sphere

Consider the two vectors to the points on the unit sphere, $${\bf v}_i=(\sin\theta_i\cos\varphi_i,\sin\theta_i\sin\varphi_i,\cos\theta_i)$$ with $i=1,2$. Use the dot product to get the angle $\psi$ ...
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71 views

Turn any shape to circle

I'm looking at trying to calculate and re position 3d vectors to align in a new position to form a circle. I've achieved this already however only when all points are evenly distributed in the same ...
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332 views

Generating evenly distributed points on a sphere

How could I write an algorithm to generate n points distributed 'evenly' on a sphere? I already wrote an algorithm to generate points distributed uniformly on the surface (here), but by 'evenly' ...
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1answer
69 views

Coordinates transformation and falling body trajectory

As it's well known, assuming the earth fixed and non rotating, the trajectory of a falling body with initial speed $v_0 = \{v_{0x},v_{0y},{v_{0z}}\}$ is contained in a plane. Assuming an observer in ...
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1answer
80 views

Getting coordinate between two coordinates knowing the distance and latitude

That is my wall: http://imageshack.us/f/266/wall2z.png/ I know the coordinates of the lower points (left and right). (X1,Y1,Z) and (X2,Y2,Z) where X es the latitude, Y longitude and Z the altitude. ...
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68 views

Shooting projectiles with GPS locations

I need a formula that given a GPS location a cannon ball will be shot from there, and I need to find the other GPS location that the cannon ball will fall. I can get direction, considering a circle ...
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205 views

MGRS 100 km x 100 km square calculation

In the MGRS (Military Grid Reference System) after being split into UTM zones, the earth is split into 100km x 100 km sections defined by two letters excluding I and O. these sections repeat and colom ...
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1answer
52 views

An error in Wikipedia? (trigonometry)

https://en.wikipedia.org/wiki/N-sphere In "Spherical coordinates" section the hyperspherical coordinates are results of arccosinus function. In some other sources there is arccotangent instead: ...
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1answer
117 views

How to use geometry to express unit vectors of spherical coordinate system in terms of Cartesian unit vectors

It's quite easy to express unit vector $\hat{r}$ in sum linear combinations of Cartesian unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$. But I am not sure how I can use geomtery to find a Cartesian ...
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1answer
226 views

Finding out the nodes from spherical symmetric equation for mathematica and then plotting it in gnu-plot

What I want to do is to draw a curve for the spherical partial differential equation for S at rho=0: \begin{equation} \frac{\partial^2S}{\partial \rho^2}+\frac{d-1}{\rho}\,\frac{\partial S}{\partial ...
-1
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1answer
172 views

Spherical coordinates to cartesian coordinates.

I want to find out the distance between the centers of $2$ circles. Say, circle $1$ $(\theta,\phi)$ circle $2$ $(\theta,\phi)$ The radius of this circle is found using $d\tan(\theta)$ where $d$ is ...
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1answer
3k views

Volume of a Cylinder Using Cylindrical Coordinates and Triple Integration

Calculate the Volume V of a right circular cylinder of radius a and height h, using cylindrical coordinates and triple integration.
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1answer
291 views

$dxdydz \to -r^2\sin(\theta)\sin(\phi+\theta)dr d\phi d\theta$?

So I got this answer $-r^2\sin \theta\sin(\phi+\theta)dr d(\phi)d(\theta)$ which I think is wrong because I googled it and it must be $-r^2\sin\theta dr d\phi d\theta,$ but $\sin(\phi+\theta$) clearly ...