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).

learn more… | top users | synonyms

0
votes
0answers
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 ...
1
vote
1answer
257 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 ...
2
votes
1answer
3k views

building transformation matrix from spherical to cartesian coordinate system

How to arrive at the following from given $ x = r\sin \theta \cos \phi, y = r\sin \theta \sin \phi, z=r\cos\theta $ $$ \begin{bmatrix} A_x\\ A_y\\ A_z \end{bmatrix} = \begin{bmatrix} \sin ...
1
vote
1answer
479 views

Coordinates of interception point Y with XY being the shortest distance of X to AB on sphere

How would one calculate the interception point $Y$ with $\overleftrightarrow{XY}$ being the shortest distance of $X$ to $\overleftrightarrow{AB}$? This answer to the question How to find the ...
1
vote
1answer
188 views

Parametric representation of rectangular form in terms of parameters $\rho$ & $\theta$

I need to represent the cone $z=\sqrt{3x^2+3y^2}$ parametrically in terms of $\rho$ and $\theta$ where $(\rho,\theta,\phi)$ are spherical coordinates. Attempt. I tried using: ...
0
votes
1answer
840 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 ...
3
votes
1answer
394 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 \ ...
5
votes
4answers
898 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 ...
0
votes
1answer
225 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 ...
2
votes
1answer
331 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
0answers
631 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 ...
1
vote
0answers
146 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 ...
1
vote
0answers
132 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 ...
3
votes
1answer
1k 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: ...
1
vote
1answer
73 views

Integral from Sphere to Disc

Suppose one has an integral of the form $\int_{S_1^{d-1}} f(\phi(v)) d \text{vol}_{S_1^{d-1}}(v)$. Here $S_1^{d-1}\subset \mathbb{R}^d$ is the unit sphere. Let $B_1^{d-1}\subset\mathbb{R}^{d-1}$ be ...
1
vote
1answer
3k 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% ...
8
votes
4answers
14k 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 ...
2
votes
2answers
460 views

Evaluating an integral in spherical coordinates over on odd shaped region.

I have to evaluate this integral: $$ \int_{0}^{\sqrt{2}}\int_{y}^{\sqrt{4-y^2}}\int_{0}^{\sqrt{4-x^2-y^2}} \sqrt{x^2+y^2+z^2}dzdxdy $$ in spherical coordinates. I see that the region in the xy ...
3
votes
0answers
794 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 ...
10
votes
1answer
752 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 ...
1
vote
4answers
682 views

Coordinates notation in Spherical polar system

There is a number of conventions for specifying coordinates in Spherical polar coordinate system: ($r$, $θ$, $φ$), ($r$, $φ$, $θ$), ($\rho$, $\theta$, $\phi$) and even ($r$, $\psi$, $θ$). The article ...
9
votes
2answers
2k 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. ...
0
votes
1answer
336 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 ...
3
votes
1answer
1k 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 ...
0
votes
0answers
206 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 ...
3
votes
2answers
756 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 ...
1
vote
1answer
2k views

Transforming from one spherical coordinate system to another

I have a set of points on the surface of a sphere specified in one coordinate system (specifically, the equatorial coordinate system), and for each point I need to work on all its neighbouring points ...
2
votes
1answer
316 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 ...
3
votes
1answer
273 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 ...
1
vote
1answer
514 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 ...
0
votes
2answers
1k views

Find the average value of this function

Find the average value of $e^{-z}$ over the ball $x^2+y^2+z^2 \leq 1$.
2
votes
1answer
315 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
4answers
1k 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.
1
vote
1answer
3k views

Line element (dl) in spherical coordinates derivation/diagram

I'm trying to figure out how to derive the formula for an infinitesimal length $\mathrm dl$, specifically, the first formula in this Wikipedia section: $$\mathrm d\mathbf r=\mathrm dr\hat{\boldsymbol ...
3
votes
4answers
1k 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 ...
4
votes
2answers
323 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 ...
2
votes
2answers
266 views

How to use a Rhumb Line?

I am new to working with coordinate data and figured out the equation I am looking for is the Rhumb Line. I went to go research it and found a lot of equations and I still have no idea where to start. ...
1
vote
1answer
650 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
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 ...
0
votes
1answer
229 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 ...
1
vote
1answer
1k 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 ...
2
votes
1answer
453 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 ...
1
vote
1answer
142 views

Find third point in mapping system

I have two points defined: $A$ and $B$ For both I know $x,y$, longitude, and latitude (gps coordinates). How do I calculate $x,y$ of a third point $C$ when I know its longitude and latitude? I know ...
4
votes
1answer
689 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 ...
3
votes
1answer
347 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 ...
1
vote
1answer
365 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\, ...
2
votes
0answers
487 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
1answer
850 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!
5
votes
4answers
1k 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 ...
5
votes
1answer
3k 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 ...