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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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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146 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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3answers
57 views

Triple Integral in Spherical Coordinates.

$\newcommand{\de}{\operatorname{d}}$A little stuck on this one. $$\iiint_V ye^{-(x^2+y^2+z^2)^2}\de V$$ Use Spherical Coordinates to evaluate where V is the solid that lies between y=0 and the ...
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1answer
54 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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1answer
60 views

Find the volume inside

Find the volume inside the torus $\rho=\sin\phi$. First of all how can $\rho=\sin\phi$ represent a torus? I can't even visualise that. All Ideas are welcome, this looks like a 'food for thought ...
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1answer
123 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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2answers
26 views

Divergence of vector in spherical coordinates

How should I calculate the divergence for $$\vec{V}=\frac {\vec{r}}{r^2}$$ Is it possible to convert it from spherical coordinates to cartesian?
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610 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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1answer
962 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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2answers
22 views

Derivation for the integrating term in line integrals and volume integrals in spherical coordinates

Can anyone refer me to, or respond with, the derivation for the integrating term in line integrals $dl=dr\hat{r}+rd\theta\hat{\theta}+r\sin\theta\ d\phi\hat{\phi}$ and volume integrals ...
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43 views

How can I solve these two tough integrals?

\begin{equation*} J_{1} = \int_{0}^{\sqrt{{\pi}/{6}}} \int_{y}^{\sqrt{{\pi}/{6}}} \cos{(x^2)}\,dx\,dy \end{equation*} \begin{equation*} J_{2} = \int\int_{E}\int z e^{(x^2+y^2)} + xe^{x^8}\,dV, ...
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380 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
619 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
100 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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1answer
352 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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1answer
377 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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1answer
80 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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1answer
50 views

convert triple integral $\int_{-6}^{6}\int_{-\sqrt{36-x^2}}^{\sqrt{36-x^2}}\int_{x^2+y^2}^{36}x\,dz\,dy\,dx$ to spherical coordinates

I need to convert the following integral from rectangular to spherical coordinates $$ \int_{-6}^{6}\int_{-\sqrt{36-x^2}}^{\sqrt{36-x^2}}\int_{x^2+y^2}^{36}x\,dz\,dy\,dx $$ I am able to convert it to ...
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1answer
23 views

Spherical coordinates on cartesian straight lines

I'm trying to solve this problem: Compute the volume of the solid bounded by: the surface $(z+1)^2=x^2+y^2,$ the surface $4z=x^2+y^2,$ above the $xy$ plane. I want to do it with spherical ...
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1answer
19 views

Envisioning Spherical Coordinates

Is there a way to envision these two equations in spherical coordinates without plotting a bunch of points? I'm interested in what the surfaces look like. $$\rho=\sin\theta\sin\phi$$ ...
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1answer
16 views

Converting to Spherical Coordinates that have a Large Azimuth?

I've run into a problem converting Cartesian coordinates to spherical coordinates. Say I've got a vector/point $p=(-1,5,7\frac{2}{3})$. Obviously, finding the polar angle/inclination isn't going to ...
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1answer
21 views

Integration in d-dimensional spherical coordinates

Can someone please tell me why $$\left(\int_0^\infty dr\, e^{-r^2}\right)^d=\int_0^\infty dr\,r^{d-1}S_d e^{-r^2}?$$ Why doesn't $d$ end up joining the exponential?
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1answer
101 views

Determine if one point lies between two other points on a sphere

My question is rather simple. Can I use the dot product to determine if a coordinate lies between two others? With coordinates I mean a Point P(latitude, longitude) on the surface of the sphere. I ...
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1answer
40 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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1answer
21 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
35 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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1answer
10 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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1answer
52 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
39 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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1answer
52 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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2answers
81 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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2answers
178 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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1answer
79 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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1answer
268 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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1answer
95 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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1answer
252 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
373 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
246 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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19 views

Finding the volume between a cone and a sphere

I have to find the volume between the sphere $x^2+y^2+z^2=1$ and below the cone $z=\sqrt{x^2+y^2}$ using Spherical Coordinates. Here is what I have so far: Transforming the cone part gives: ...
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14 views

Parameterization of a closed curve on a sphere

I'm looking for a parameterization of a closed curve C on a sphere. assume the projections of C on y-z, x-z, x-y plane are f(x), g(y), h(z), respectively, and ${\oint}f(x)dx={\oint}g(y)dy=0$, and ...
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19 views

Rotation in spherical coordinates (w/o Cartesian)

What are the rotation matrices in polar coordinates? Which matrices I should multiply by a unit vector $(1, \theta, \phi)$ to rotate the latter around basic axes to angle $\beta$? Used notation: ...
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1answer
40 views

Rotation matrix in spherical coordinates

When given arbitrary point on a unit sphere $a = (\theta, \phi)$ and an arbitrary axis $\vec{A}=(\Theta, \Phi)$, can we have an algebraic expression for $a_1=(\theta_1, \phi_1)$ which is a rotation of ...
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14 views

Stereographic Projection preserves angles at the south pole

Show that stereographic projection preserves angles at the "south pole" $S=(0,0,-1)$. I really don't know how to approach this problem. The main problem is that I do not have a good definition ...
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30 views

Loxodrome : found an error on wolfram MathWorld web site?

Could it be?... We find this claim on Wolfram MathWorld site http://mathworld.wolfram.com/SphericalSpiral.html The claim is that this curve (given in oblate spheroidal coordinates in the limit where ...
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43 views

Find a point C on line segment AB such that line segment DC is perpendicular to AB. D is a point outside the line segent

Find a point C on line segment AB such that line segment DC is perpendicular to AB. D is a point outside the line segment. Note, Point A,B and C are in latitude,longitude format, i.e A = {lat,long} ...
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1answer
42 views

Loxodrome parametric equations

I have been trying to understand HOW one arrives at the equations $x=cos(t)cos(c)$ $y=sin(t)cos(c)$ $z=−sin(c)$ of the loxodrome. I can see that if the transformation to spherical coordinates is ...
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27 views

2D histogram from points on spherical surface of spherical coordinate system

What are the options available for creating a 2D histogram of points distribution on a sphere? Both polar angle (phi) and azimuth angle (theta) are known, and radial distance is 1.
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20 views

Rotation invariant method to compare points on spherical surface

Is there any rotation invariant method that I can use to compare the similarity between the three groups of "A" points as shown below?
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20 views

Intermediate vector naming convention when converting 3D Vector to Spherical Coord

Given the "Red" arrow as the vector of interest in the image below, is there a standard (or practical) name for the "Blue" vector? This is for the purpose of naming variables/methods related to the ...
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1answer
321 views

Rotating a point in spherical coordinates around Cartesian axis

If I have a point in spherical coordinates, and I rotate it around one of the Cartesian axes, what will be the new spherical coordinates for the point? Both spherical and Cartesian coordinate systems ...