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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Conversion from Spherical Coordinates to Cartesian Coordinates aligned along arbitrary polar axis

I have spherical coordinates $w = (\theta, \phi)$ such that $\theta$ is the angle between $w$ and the polar axis (let's assume $z$ is up). Assuming $w$ is a unit vector, the conversion to cartesian ...
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1answer
62 views

How to derive curl in spherical coordinates

This is one of those questions where I know I am making a dumb mistake someplace and I am trying to check where it is. $$ \begin{vmatrix} \frac{\bf\hat r}{r^2\sin\theta} & ...
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105 views

Angled Spherical Sector - formulas?

I am glad to be here. First off, please excuse my almost saddening lack of knowledge. Math (in general) isn't my strong point. I might ask some really basic questions, you have been warned :) The ...
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1answer
498 views

Reverse use of Haversine formula

Alright the title is not the best. What I want to do is to change the given parameters in Haversine's formula. If we know the lat,lng of two points we can calculate their distance. I found the ...
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65 views

Joint PDF for spherical region

A sphere has a coordinate system (r, $\theta$, $\phi$) with the origin at the center of the sphere. What is the joint PDF of the r and $\phi$ coordinates, $f_{r,\phi}(r,\phi)$, for a randomly ...
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203 views

Spherical coordinates of a unit vector around a normal $N$

So if I have a unit normal for a surface $N(x,y,z)$ and an incident unit vector $V(x,y,z)$ to that surface, how would I represent the vector V in spherical coordinates relative to the normal?
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151 views

Directions in spherical coordinates

Say I have a system with standard spherical coordinates. There's a man on that sphere and he's standing on the equator facing east. He chooses a random angle $0°-360°$ and turns that much in the clock ...
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308 views

Distance of two points in spherical coordinates

Today I was trying to solve a physics exercise and ran into some mathematical problems. Consider two concentric spheres $K_{1,2}$ with radii $R_{1,2}$. I wanna solve the following integral: $$E_{ij} ...
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155 views

Centre of a spherical triangle

Suppose I have a triangle defined by 3 unit vectors {$v_1, v_2, v_3$} in a 3 dimensional complex inner product space. What would be the centre of such a triangle? I guess it should be something like ...
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484 views

Rotating co-ordinates in 3D

Suppose I have 3 axes, $x$, $y$, and $z$ such that $x$ is horizontal, $y$ is vertical, and $z$ goes in/out of the computer screen where $+$ve values stick out and $-$ve values are sunken in. Suppose ...
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65 views

Knowing coordinates of a point having two coordinates and the distance.

I have the two geographic coordinates of the lower corners of a wall. So, for example, i want to know what is the coordinate that is for example 15cm on the right of the lower corner left. Is that ...
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155 views

Cross-section of a circle with a three-dimensional Gaussian

Suppose I have a three-dimensional Gaussian with mean $\bar{\mu}$, volume $A$ and covariance matrix $\Sigma$ $$G(X)=\frac{A}{\sqrt{(2\pi)^{3}\det(\Sigma)}}e^{-\frac{1}{2}(X-\mu)^{T}\cdot ...
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1answer
252 views

dotting gradient in spherical coordinates with displacement vector

The gradient in spherical coordinates is given by: $\nabla f = \left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial ...
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142 views

Relative spherical coordinates of 2 points.

I have 2 points in space, defined by their spherical coordinates. I'd like to know the spherical coordinates of the second point in a reference system centered on the first point (I know the unit ...
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130 views

What is the name for the axis that “inclination angle” is measured from in spherical coordinates?

What is the name for the axis that "inclination angle" is measured from in spherical coordinates? I keep trying to call it "polar axis", and the other axis from which the azimuth is measured, the ...
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1answer
2k views

Can you not rotate spherical coordinates?

I have some points that sit on the hemisphere in spherical coordinates: $\theta \in [0,\pi/2]$, $\phi \in [0, 2\pi]$ (ie so a hemisphere around the vector (1,0,0) (spherical coordinates). I should ...
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1answer
34 views

Granted I have NE and SW coordinates for a rectangle, how do I get the center point?

I've got the NE and SW coordinates/points for a minimum bounding rectangle. How do I calculate the center point of this rectangle? At first thought, I could calculate this using simple division. ...
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1k views

Find the average value of this function

Find the average value of $e^{-z}$ over the ball $x^2+y^2+z^2 \leq 1$.
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110 views

Finding the limits of a triple integral.

Evaluate: $$\iiint_V (x^2+y^2)\:\mathrm{d}x\:\mathrm{d}y\:\mathrm{d}z,$$ where $V$ is the region in the positive octant bounded by the sphere $\|\vec{r}\|=a$. I am unsure how to get the limits of ...
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477 views

Identifying a surface $\rho^2\cos(2\phi)-1=0$

I need convert this spherical expression, to a rectangular form (specific surface). $$\rho^2\cos(2\phi)-1=0$$ Thanks for a while.
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832 views

Converting from Spherical to Rectangular

I need to convert $\rho \sin\phi=2\cos\theta$ in to rectangular form. Attempt: I tried using those nice properties : $$x=\rho\sin\phi\cos\theta \\y=\rho\sin\phi\sin\theta\\z=\rho\cos\phi$$ and ...
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255 views

How to move a camera in “Flight Simulator” style (Roll, Pitch, Yaw) using a joystick

I am working on a video game where the camera movement (what the viewer sees) is controlled by a joystick. I want the camera movement to act like a flight simulator meaning the following: When the ...
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2answers
498 views

How to calculate the angle between two vectors, defined by 3 points on the earth?

I want to develop a formula to calculate the angle between two vectors. The vectors will be OX and OY (from point O to X , and Y), where the points are defined by their latitude and longitude values. ...
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1answer
73 views

Exterior Product $d\Phi_1\wedge d\Phi_2$ and spherical coordinates

One short question: If $\Phi\colon\mathbb{R}^3\to\mathbb{R}^3$, defined by $$ \begin{pmatrix}r\\\vartheta\\\phi\end{pmatrix}\mapsto\begin{pmatrix}r\sin \vartheta\cos \phi\\r\sin \vartheta ...
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238 views

Triple Integral Spherical Coordinates

So I have to compute the triple integral of this: $\int\int\int \frac{1}{1+x^2+y^2+z^2}$ and it says the equation of the sphere is $ x^2 + y^2 + z^2 = z$ which is just an elongated sphere running ...
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76 views

Spherical Coordinates

Let Q be the region above by the plane $8z=4-x-y$ and below by the cone $64z^2=x^2+y^2$. How would I setup the triple integral to find volume of Q, using spherical coordinates? I just need help with ...
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37 views

How do I find the limits for $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$?

Evaluate $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$ where $W$ is the solid bounded by the two spheres $x^2 + y^2 + z^2 = a^2$ and $x^2 + y^2 + z^2 = b^2$ where $0 < b < a$. ...
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17 views

Conversion to Spherical Coordinates

http://www.physics.usu.edu/Wheeler/QuantumMechanics/QMOrbitalAngularMomentum.pdf Unfortunately, this is the first time I've come across a conversion to spherical coordinates and I'm pretty lost. The ...
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1answer
25 views

3D Fourier transforms of $e^{-\beta r} $ and $re^{-\beta r} $

I am trying to find the integrals $$\large\int\limits_{\mathbb{R}^3} e^{-\beta \left|\vec{r}\,\right|}e^{i \vec{q} \cdot\,\vec{r}} \mathop{d^3r}$$ $$\large\int\limits_{\mathbb{R}^3} ...
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8 views

Uniform sampling from part of sphere surface

I'd like to pose a question about uniform sampling on the surface of a sphere. I searched this site, and uniform sampling on a sphere surface seems to be quite a common problem. The common solution ...
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32 views

Trouble with understanding a spherical coordinate system.

We have a sphere with $r=1$, and we want the coordinates of $C$. $A$ is the north pole, and $AB$ is our prime meridian. See picture: I'm familiar with an $(x,y,z)$ coordinate system, but not so ...
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1answer
33 views

Curving points to a sphere

I think math.stackexchange is the right place to post this, but if not, feel free to tell me. I have a series of points to be plotted on a sphere (Each one has a latitude and longitude value). These ...
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44 views

Spherical Coordinates Representation

I just wanted to know what the set of all points in which spherical coordinates can be shown in more than one way is? I think it is only the origin but I am not sure
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80 views

Finding the the radius of a sphere

I'm having a hard time to find the radius of this sphere equation. I got the center correct, but I can't get the correct answer for the radius. I'm completing the square, but my solution is off. I ...
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144 views

Spherically Symmetric Function

Suppose $f:\mathbb{R}^3\setminus B(0,1) \to \mathbb{R}$ is smooth and satisfies $f(S^2)=0$, i.e. the unit sphere is a level set of $f$. Does it necessarily follow that $f$ is a spherically symmetric ...
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252 views

Finding a third coordinate on a sphere that is equidistant from two known coordinates

Here is my problem that I'm having some trouble with: I have the coordinates (latitude and longitude) of two points on Earth. I have no problem finding the great circle distance between the two ...
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73 views

How to minimize the length of a curve on $S^2$

The length of a curve $\gamma$ starting from a point $p$ and ending at another point $q$ on $S^2$ is given by the formula $$l_{\gamma}(S^2)=\int_{0}^{1}\sqrt{(d\phi/dt)^2+ \sin^2\phi ...
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237 views

Given an arc length and an angle, how do I get a sphere coordinate?

Assuming I start at the top of a sphere and am given the radius of the sphere, an angle to turn, and a distance to walk along the sphere, how could I find my destination in the sphere coordinate ...
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93 views

Transformation of coordinates

Given a point P with spherical coordinates $(r_p, \phi_p, \theta_p)$ on the sphere: $$(x-a)^2 +(y-b)^2 +(z-c)^2 = R^2$$ and a line through the center of the sphere with equation : $x=a+\alpha$ , ...
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222 views

Relatively simple geometry with latitude/longitude coordinates

I'm trying to calculate a user's deviation from a route defined as a list of latitude/longitude points that make up the route. My proposed solution is to try and calculate the distance between the ...
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1answer
335 views

Finding the sign of $\phi$ in spherical coordinates

I know its a little silly, but I got the wrong sign several times. Just to be clear, $z=r\cos(\phi), -\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$ when converting from cartesian to spherical. So, how do I ...
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1answer
228 views

Where can I find specific Jacobi determinants in the Bronstein-Semendjajew reference work?

I'm trying to find the fact that the Jacobi determinant (functional determinant) of the cartesian->spherical coordinate change is $r^2 \sin\theta$ in a mathematical reference book, "Taschenbuch der ...
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From points on a latitude-longitude map to direction on hemisphere

I've wrote a program in matlab that following a given algorithm individuate some points on a latitude-longitude map. This is an example output: Now I need to map these points (in blue) to ...
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Transforming uncertainties in spherical coordinates

I'm not looking for a very rigorous answer, it's a practical problem that I've encountered and whichever solution is best will be determined by trying it out on my data. I have a measurement of an ...
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15 views

Unit vectors in spherical co-ordinate system

I see that a vector can be described in spherical co-ordinates, with respect to it's cartesian co-ordinates as $$ ...
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2answers
58 views

converting kph and heading to xyz velocity vector

I am writing software (in C++) that is required to send out messages from our simulation system to another simulation system. Problem is we track the simulation object's current speed (kph) and ...
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16 views

Interesting question about a measurement using $J^2$

I really dont understand how to do part d)iv) on this question. This seems strange as it is only worth 2 marks? What step am I missing, I feel this may be rather obvious to others. for part d)i) I ...
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1answer
40 views

N points in a circle around a point on a sphere.

Consider a 3D sphere: $(x_{c}, y_{c}, z_{c})$ : cartesian coordinates of the center $r$ : the radius Consider a random point on the surface of this sphere of coordinates : $(x_{0}, y_{0}, ...
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31 views

grouping geographical points in spherical bins

I want to create spherical bins in the Earth's surface which will encompass already distributed geographical points for subsequent purposes. The two requirements are that the spherical bins should ...
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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 ...