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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Radius of the Earth at N32.704220, W90.000000?

I want to express a point on a map in radian spherical coordinates. By Google maps, this location is north of Canton, MS, USA just a few hundred feet from US 51. In radian spherical coordinates, ...
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Spherical distance

The spherical distance between two points (P1=(0,0,1) P2=($\frac{1}{2\sqrt{2}}$,$\frac{1}{2\sqrt{2}}$,$-\frac{\sqrt{3}}{2}$) ) is $\frac{5\pi }{6}$ I am at a loss as to how the spherical distance was ...
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26 views

Writing same equation in different forms

I am working with a unit circle with imaginary integration. I know from experience that this can be written as $f(\theta)=\cos t+ i \sin t$ or $e^{i \theta } $ My question would be if i have a circle ...
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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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27 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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22 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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28 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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27 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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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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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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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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17 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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69 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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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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60 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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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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222 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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84 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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305 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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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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266 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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438 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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261 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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Proving that the Jacobian for spherical coordinates and the Jacobian for integrating over a sphere coincide

I want to use the formula for switching an integration from spherical to cartesian coordinates : $\int_{RS^{n-1}} \! f(x) \, \mathrm{d}\sigma (x)$=$\int\int...\int ...
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volume using spherical coordinates

Let $$V = \{(x, y, z): x^2 + y^2 ≤ 4 , 0 ≤ z ≤ 4\}$$ be a cylinder and let $P$ be the plane through $(4, 0, 2), (0, 4, 2)$ and $(−4, −4, 4)$. Compute the volume of $C$ below the plane $P$. I'm having ...
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27 views

Solving LES containg spherical coordinates

i have a three-dimensional parametric equation of a line, where the directional vector is normalized and converted to spherical coordinates to calculate a angle offset. It looks like this: ...
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11 views

Comparing paired objects in 3d space using direction angles.

I have data about paired objects in 3d space, where each object is defined by three components. What I would like to do is compare the orientations of the paired objects and see if one group of those ...
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Midpoint of two n-vectors

Currently I am using n-vectors (more information can be found here) for an accurate and singularity-free representation of coordinates on the earth's surface. For various reasons, I need to compute ...
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Normal form of plane

I propose a simple equation of plane generalized from 2D Normal form of straight line as follows, starting from Straight line Normal form: $$ x\, \cos \alpha + y \sin \alpha = p. \tag{1}$$ ...
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Heat Equation: Remains finite?

I am doing a PDE question. It's about heat equation, spherical coordinates (the usual stuff). The boundary condition is $\frac {\partial T}{\partial r} (1,t) = 0 $ and it also said for $T$ to remain ...
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28 views

Short question about spherical coordinates

If I have a vector orthogonal to the $x$-$y$ plane of an $xyz$ axis system, I mean, a vector with just $z$ component: How can I express it in spherical coordinates?
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Setting up triple integral in spherical coordinates

Integrate $f(x, y, z) = z^2$ over $A = \{(x, y, z) \in \mathbb{R^3} | x^2+y^2+z^2 \leq R^2, x^2 + y^2 + z^2 \leq 2RZ\}$ I know $A$ is the intersection between two spheres but I am unable to figure ...
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Translation matrix in spherical coordinates system

I'm using WorldWind software to draw segments (polyline) on the globe to materialize an aircraft flightplan. Each point in a flightplan is named waypoint. Waypoints are expressed in geographical ...
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42 views

sphere-sphere intersection

Let $ S_1 : (x-1)^2 +y^2+z^2=1 $ $S_2 : x^2 +y^2 +z^2 =1$ $S_3 : (x+1)^2 +y^2 +z^2 =1 $ Find the volume of the solid inside $S_2$ and outside $S_1$ and $S_3$, using triple integrals. I have try ...
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20 views

x1, y1 and radius are given - can anything be assumed about x2, y2?

I have a list of lat/lng coordinates. Given the coordinates x1, y1, and a radius r -- is there anything I can assume about the coordinates that fall within the radius of x1, y1? For example, can I ...
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Does anybody know how to actually derive spherical harmonics in a way that is historically accurate and intuitive?

And by "historically accurate", I mean without resorting to techniques of derivation which were developed after the fact or explanations which use the very concept they're trying to explain. The few ...
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1answer
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Find geo coordinate by a coordinate and an angle

I need some help with this problem. I have a GPS coordinate and an angle in degrees. I need a new GPS coordinate x km away from the point I already have. Degree is counted clockwise and y-axis is ...
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Parametrization of sphere including constant inclination $(\theta, i)$ geodesics

Find parametrization of sphere with respect to $\theta$ = constant meridians and i = constant inclination geodesic circles passing through N-S axis and E-W axis respectively. The Earth does not rotate ...
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36 views

Cartesian to Spherical Coordinate Conversion for Triple Integral

I have a question regarding what happens to the boundaries when converting a triple integral from Cartesian to Spherical Coordinates. Example ...
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Solutions of the Laplace Equation in spherical coordinates

I would like some help with the following problem. Thanks for any help in advance. Use spherical coordinates to find all solutions of the Laplace equation ∆u(x)=0, u∈Ω⊂R3 that depend only on the ...
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$\delta$ in spherical coordinates: $\int_0^R\int_0^{2\pi}\int_0^{\pi}\delta(\theta-\pi/2)(r^2\sin(\theta)\,d\theta \,d\phi \,dr)$

Suppose you have a disc of radius $R$, we can find its area in polar coordinates by: $$\int_0^R\int_0^{2\pi}(r\,d\phi \,dr)=\pi r^2$$ Naively, I also expect to be able to integrate in spherical ...
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30 views

Intensity distribution of a Lambertian LED as a function of angle

I have a practical spherical geometry problem that I'm having trouble cracking. I'm illuminating a planar surface with an LED that has a Lambertian intensity distribution, i.e. the intensity drops off ...
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120 views

Finding the (unit) direction vector given azimuth and elevation

I want to calculate a unit direction vector of a direction with given the azimuth and elevation (cf. http://en.wikipedia.org/wiki/Azimuth), respectively $$\alpha \in [0^{\circ},360^{\circ}), \qquad ...
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117 views

Mapping Coordinates on a Plane Tangent to a Sphere into Cartesian Coordinates in 3D Space

Before we begin, I must ask you to keep the vocabulary at high school level. These variables define the point the plane needs to be tangent to – the center of the circle is at the origin. r - defines ...
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Can someone help me convert definitions of hip movement in Cartesian coordinates into spherical coordinates?

I am a biomechanist. I am having a problem converting an idea in Cartesian coordinates $(x,y,z)$ to spherical coordinates $(r,\theta,\phi)$. I wondered if someone could help me. I can physically ...
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Extract Spherical Harmonics from integral

In physics we may find integrals in the format $ I = \int \mathrm{d}b \, b^2 F(b) \mathrm{d}\Omega' \frac{Y_l^m(\theta',\phi')}{|{\bf a} - {\bf b}|^2} $ where the vector ${\bf a}$ has spherical ...
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Transform Confocal Ellipsodal to Spherical Coordinates

I heard that someone published a paper showing that the confocal ellipsoidal coordinate system can transform into the spherical coordinates under special limit evaluations, however I was unable to ...
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25 views

Cartesian to spherical coordinate system

Hey I want to convert Cartesian to spherical coordinate system. I referred many site and for calculating elevation angle $\theta$ from positive z axis they all used formula $\arctan \frac { ...
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Is the spherical harmonic representation of a 2D field independent of grid?

What I am currently unable to understand is whether the spherical harmonic representation of a 2D field is in any way tied to the nature of the grid on which decomposition/composition is performed. I ...
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63 views

Spherical coordinates and the Law of Cosines

I have one question on my project. I am assuming earth is a perfect sphere. How can I get from the Law of Cosines $$\cos(c)=\cos(\operatorname{lat} A)\cos(\operatorname{lat} ...