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).
9
votes
1answer
415 views
Can someone please explain the cube to sphere mapping formula to me?
I am wondering if anyone could explain how the following formula works, it is supposed to take the input as a point on a cube then map that to points on a sphere, please go gentle on me, I'm in 9th ...
7
votes
1answer
2k views
How to perform a Fourier transform in spherical coordinates?
For a function $f(r, \vartheta, \varphi)$ given in spherical coordinates, how can the Fourier transform be calculated best? Possible ideas:
express $(r,\vartheta,\varphi)$ in cartesian coordinates, ...
4
votes
4answers
761 views
How to find the distance between a point and line joining two points on a sphere?
How do I calculate the distance between the line joining the two points on a spherical surface and another point on same surface? I have illustrated my problem in the image below.
In the above ...
4
votes
4answers
571 views
Why is the differential solid angle have a $\sin\theta$ term in integration in spherical coordinates?
When you integrate in spherical coordinates, the differential element isn't just $ d\theta d\phi $. No. It's $\sin\theta d\theta d\phi$, where $\theta$ is the inclination angle and $\phi$ is the ...
4
votes
2answers
170 views
How do I convert a vector field in Cartesian coordinates to spherical coordinates?
I have a vector field in terms of $\mathbf{\hat i}$, $\mathbf{\hat j}$, and $\mathbf{\hat k}$,
$$\mathbf{F} = x\mathbf{\hat i} + y\mathbf{\hat j} + z\mathbf{\hat k}$$
How do I convert it to the ...
4
votes
4answers
340 views
What is the geodesic between a point and a line (geodesic between two points) on an oblate spheroid?
I found a similiar question that also asks for the distance from a point to a line but works on a sphere.
Now I'm trying to figure out the length of the geodesic line ...
4
votes
1answer
472 views
How to integrate by parts in spherical coordinates
I'm running into some troubles with the integration of a spherically symmetric 3D function.
I'm having the following expression to evaluate :
$$
I=\int_0^{2\pi} d\phi \int_0^\pi \sin\theta ...
4
votes
2answers
53 views
derivatives transformation
I'm currently doing a calculation for the connection coefficients using the standard space-time coordinates, namely x[0],x[1],x[2],x[3]. The setup is a spherically symmetric problem.
In my ...
3
votes
2answers
84 views
Calculating longitude degrees from distance?
I need to calculate how many longitude degrees a certain distance from a point are, with the latitude held constant. Here's an illustration:
Here x represents the longitude degrees, the new point ...
3
votes
2answers
1k views
How can I pick a random point on the surface of a sphere with equal distribution?
I've got a random number generator that yields values between 0 and 1, and I'd like to use it to select a random point on the surface of a sphere where all points on the sphere are equally likely.
...
3
votes
3answers
4k views
Surface Element in Spherical Coordinates
In spherical polars,
$$x=r\cos(\phi)\sin(\theta)$$
$$y=r\sin(\phi)\sin(\theta)$$
$$z=r\cos(\theta)$$
I want to work out an integral over the surface of a sphere - ie $r$ constant. I'm able to derive ...
3
votes
1answer
37 views
problem or doubt regarding visualizing angles of spherical triangle
I must confess that I am not able to visualize or understand what is the angle of a spherical triangle say $ABC$ where $A,B,C$ are vertices of the triangle which is formed by intersection of three ...
3
votes
1answer
84 views
Integration on a sphere
I have an integral at hand which has the form of
$$I = \int_{u\in \mathbb{S}^2} f(\mathbf{u}\cdot \mathbf{s}_1) f(\mathbf{u}\cdot \mathbf{s}_2) d\mathbf{u}$$
where $\mathbb{S}^2$ is the unit sphere ...
3
votes
1answer
712 views
Projection of Gaussian in Spherical Coordinates
Consider a point with spherical coordinates $\vec{r}_0=(r_0, \theta_0, 0)$. The spherical gaussian distribution centered at $\vec{r}_0$ is $f(\vec{r})=Ne^{|\vec{r}-\vec{r}_0|^2/A}$, where $N$ is the ...
3
votes
1answer
223 views
Monte carlo integration in spherical coordinates
I was playing around with writing a code for Montecarlo integration of a function defined in spherical coordinates. As a first simple rapid test I decided to write a test code to obtain the solid ...
3
votes
0answers
152 views
Stokes' and Divergence Theorem Problems
I have 2 questions on stokes and divergence theorem each. I think I have done both and I just want to make sure that I did them correctly.
Question 1
Let C be the boundary of the surface ...
3
votes
0answers
655 views
Spherical coordinates grad and div.
Struggling with the following:
Prove the identity
$$ \nabla = e_{r}(e_{r} \cdot \nabla) + e_{\theta}(e_{\theta} \cdot \nabla) + e_{\phi}(e_{\phi} \cdot \nabla).$$
Given the vector fields ...
3
votes
2answers
390 views
Lat/Long grid points covered by projecting rectangle onto sphere
Before my question proper, a little background: I'm wanting to optimise some computer rendering by eliminating the drawing of things that aren't visible given the current view.
Suppose we have a ...
2
votes
4answers
409 views
Latitude and longitude of points on a line
How could you get the latitude and longitude of four points (equal distance apart) on a line from $(27,-82)$ to $(28,-81)$? The four points should split the line into 5 parts.
2
votes
1answer
89 views
Triple Integral in Spherical Co-ordinates
Find the volume bounded by the surface $(x^2 + y^2 + z^2)^2 = 2z(x^2 + y^2)$
I have $x = \rho \sin\phi \cos\theta$, $y = \rho \sin\phi \sin\theta$, $z = \rho \cos\phi$.
Therefore, $(x^2 + y^2 + ...
2
votes
2answers
392 views
Converting from Cartesian coordinates to Spherical coordinates
I want to understand how to convert from Cartesian coordinates to spherical coordinates. I have the following definitions:
\begin{align} x & =r\sin\theta\cos\phi \\[6pt]
y & ...
2
votes
1answer
909 views
Angle between GPS coordinates
I realize GPS Coordinates are spherical coordinates. However I know the earth is more of an ellipsoid. I need to compute with a fairly high degree of accuracy the pitch and yaw between two objects ...
2
votes
1answer
177 views
Equation for the sensitivity pattern of a bi-directional microphone?
Can anyone give me an equation that expresses the sensitivity pattern of a bi-directional microphone, as a function of azimuth and elevation angle? A bi-directional microphone pattern looks something ...
2
votes
1answer
173 views
What am I actually doing when I integrate using spherical coordinates in $\mathbb{R}^3$?
When learning vector fields and using Green's Theorem with the Jacobian to find the area of a level surface, I actually realized that most of the examples shown in my book would be much easier to ...
2
votes
1answer
210 views
Can you formulate a $ \phi, \theta $ restriction in spherical coordinates for a great circle?
Further to this question
Quaternion rotation has a nice property that you can trace any great circle you like. You specify the axis of rotation, and you will automatically follow the great circle ...
2
votes
1answer
263 views
integral of a spherically symmetric 3-dimensional function over all space
I'm very sorry because it may be a very basic question but I'm not able whether to solve it for sure, nor to find an answer in stackexchange or elsewhere.
I have to calculate
$ \int \int ...
2
votes
1answer
1k views
Vector sum in spherical coordinates
I can't seem to come up with a simple formula to head-tail adding two vectors in spherical coordinates. So I'd like to know:
Can anybody point out a way to do it in spherical coordinates (without ...
2
votes
3answers
127 views
Integration with Spherical Coordinates
Use spherical coordinates to find the volume of the solid inside both $x^2+y^2+z^2=16$ and $z=(x^2+y^2)^{1/2}$.
2
votes
2answers
370 views
Projection of a 3D spherical distribution function in to a 2D cartesian plane
Consider a 3D spherical Gaussian distribution function that depends on radius only,
$$f(r) = \frac{1}{N} e^{-(\frac{r-R_\mu}{\sigma})^2}$$
where $R_\mu$ is the radial offset of the distribution and ...
2
votes
1answer
198 views
Change of coordinate system on a sphere
This might take a while to explain, so bear with me:
I've got a perfect sphere. I've set up an arbitrary longitude/latitude ("angle") coordinate system on it (imagine an equator around the middle, ...
2
votes
1answer
336 views
Three Dimensional Fourier Transform of Radial Function without Bessel and Neumann
I am trying to compute the Fourier transform of $\frac1{|\mathbf{x}|^2+1}$ where $\mathbf{x}\in\mathbb{R}^3$.
Just writing out the integral: ...
2
votes
1answer
25 views
Determining new coordinates after a rotation of a sphere
Imagine that I am standing at a place on Earth, using coordinates of say N41 W74. Now the Earth's axis rolls 90 degrees, causing the N/S axis to become the equator, and rotation resumes as before. ...
2
votes
0answers
168 views
Inverse Jacobian matrix of spherical coordinates
I found inverse transformation from spherical coordinates to cartesian coordinates (on $x>0$, $y>0$ and $z>0$). I have
$$
r = w_1(x,y,z) = \sqrt{x^2+y^2+z^2}
$$
$$
\theta = w_2(x,y,z) = ...
2
votes
1answer
251 views
Coordinate transformation
I have some problems with a geometrical calculation.
I want to know the coordinates of the point $P_2$ in my coordinate system $A \ (x,y,z)$ as shown in the following figure.
Point $P_1$ (in $A \ ...
2
votes
0answers
337 views
Calculate the volume between two spheres
I have to find the volume between the two spheres. Their equations are
$x^2+y^2+z^2=2$ and $x^2+y^2+(z-\frac1 2)^2=\frac 1 4$ for $z>0$
The first one has center $(0,0,0)$ with $r=\sqrt 2$ and the ...
2
votes
0answers
380 views
How do I find the inverse Fourier transform of a function that is separable into a radial and an angular part?
I need to take the inverse Fourier transform of a function that is initially specified in spherical coordinates:
$f(r, \theta, \phi) = \int_{R^3}F(k, ...
1
vote
4answers
512 views
How can I get a square starting with a latitude and longitude point?
Is there a formula that given one latitude/longitude point and a radius (in kilometers ideally) would give me a square around the center?
A picture is worth some quantity of words:
I have the ...
1
vote
1answer
311 views
Optimal distribution of points over the surface of a sphere
How can one generate a distribution of N points over the surface of a sphere so that the all N voronoi cells have the same area? Which is the best algorithm for this?
1
vote
1answer
681 views
How do I find the Fourier transform of a function that is separable into a radial and an angular part?
how do I find the Fourier transform of a function that is separable into a radial and an angular part:
$f(r, \theta, \phi)=R(r)A(\theta, \phi)$ ?
Thanks in advance for any answers!
1
vote
2answers
2k views
What is the general formula for calculating dot and cross products in spherical coordinates?
I was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question ...
1
vote
1answer
1k views
Converting Lat/Long coords to Cartesian X/Y, then calculating shortest distance between point & line segment
I'm having an issue with accuracy when converting Lat/Long coordinates to X,Y and then finding the shortest distance from a Point to a Line with said coordinates.
The distance is off by around 40-50% ...
1
vote
1answer
41 views
Spherical harmonics expansion for a particular function
On the unit sphere, each square-integrable function can be expanded as a linear combination of spherical harmonics :
$$ f(\theta,\phi) = \Sigma_{l=0}^\infty \Sigma_{m=-l}^{+l} f_{lm} Y_{lm} ...
1
vote
1answer
50 views
How to integrate a vector function in spherical coordinates?
How to integrate a vector function in spherical coordinates?
In my specific case, it's an electric field on the axis of charged ring (see image below), the integral is pretty easy, but I don't ...
1
vote
1answer
76 views
full hessian, spherical coordinates
The question itself is pretty simple. I am running into confusion. Seems like there is a typo in the book. I wanna check myself. Maybe I am doing something wrong.
Suppose we have the function (which ...
1
vote
1answer
641 views
Great arc distance between two points on a unit sphere
Suppose I have two points on a unit sphere whose spherical coordinates are $(\theta_1, \varphi_1)$ and $(\theta_2, \varphi_2)$. What is the great arc distance between these two points?
I found ...
1
vote
1answer
348 views
projection of sphere on the circumscribed cube
Suppose I have a sphere. Inside the sphere I have an inscribed cube. What I am interested in is finding out what is the latitude and longitude (or coordinates) of a point on the sphere which will be ...
1
vote
1answer
574 views
Plotting a star's position on a 2D map
First off, let me say that I'm not a Mathematical wizard. Be easy on me :)
I am looking to draw a subsection of space using OpenGL on the iPhone. I have a star's RA and Dec and am needing to somehow ...
1
vote
1answer
263 views
What is the generalization of Parseval's theorem into spherical coordinates?
what is the relationship between the total power of a function given in spherical coordinates in the Fourier domain:
$E_k=\int_{\mathbb{R}^3}|F(k,\Theta,\Phi)|^2k^2 \sin(\Theta)\,dk\,d\Theta\, ...
1
vote
1answer
48 views
How to resolve this equation to another value?
Sorry guys, I don't know how to be more specific in the question without writing a way too long question...
Anyway my problem:
I have this formula to calculate the distance between two points on the ...
1
vote
2answers
52 views
Parametric Equation for Great Circle
So I've been doing a lot of searching and haven't found exactly what I'm looking for. My math skills are a bit rusty, so I haven't had luck deriving this on my own.
What I'm looking for is an ...
