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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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!
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2answers
150 views

Volume of a cube in spherical polars

Let us calculate the volume of the cube using spherical coordinates. The cube has side-length $a$, and we will centre it on the origin of the coordinates. Denote elevation angle by $\theta$, and the ...
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3answers
2k views

Find volume between two spheres using cylindrical & spherical coordinates

I've got two spheres, one of which is the other sphere just shifted, and I'm trying to find the volume of the shared region. The spheres are $x^2 + y^2 +z^2 = 1$ and $x^2 + y^2 +(z-1)^2 = 1$ I know ...
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73 views

Evaluate the integral $\iiint\limits_E x^2 \,\, \mathrm{d}V$

Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$. This is an exercise from my professor guide. What I tried so far: These exercise ...
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2answers
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Different ways for calculating distance between two geodetic points give me different results

I'm trying to calculate the distance between two geodetic points in two different ways. The points are: A:(41.466138, 15.547839) B:(41.467216, 15.547025) The ...
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995 views

Deriving equations of motion in spherical coordinates

OK, we've been asked to derive the equations of motion in spherical coordinates. According to the assignment, we should end up with this: $$ \bf \vec{v} \rm = \frac{d \bf \vec{r} \rm}{dt} = \dot{r} ...
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264 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 $z=(x^2+y^2)^{1/2}$.
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175 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 ...
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1answer
55 views

Inclusion, pullback of differential form

Let $\omega=x\,dy\wedge dz +y\,dz\wedge dx+z\,dx\wedge dy$ or in spherical coordinates (unless I had made some mistake) $\omega=r^3\cos \theta\, d\phi\wedge d\theta$. Now I want to find $i^*\omega$ ...
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61 views

Conversion from Cartesian to spherical coordinate for vectors - ray tracing application

I'm implementing a ray tracer that support physically based rendering, so is based on various BRDF models. At the moment I'm focused on Oren-Nayar and Torrance-Sparrow model. Each one of these is ...
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1answer
66 views

Energy of particles on sphere and uniform rotation

I have a computer program containing some particles on a unit sphere, characterized by their positions $\{(\theta_i,\phi_i)\}$. They have a total energy given by the arc distance between particle ...
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180 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 ...
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2answers
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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 & ...
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1answer
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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 ...
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529 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 \ ...
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337 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 ...
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510 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 ...
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2answers
53 views

substituting spherical coordinates to evaluate an integral.

I have to evaluate $$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{-\sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}} \, dz \, dy \, dx$$ using spherical coordinates. This is what I have come up with \begin{align} ...
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Determine a point lie in bisect area between 2 circles on sphere

Given 2 points A,B,O on sphere of radius $R$. Point O is in middle of AB. E and F are deviation from O by geodesic distance $d$ (angle between EF and AB is $90^o$). Consider 2 circles $C_1,C_2$ on ...
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1answer
78 views

Surface integral of a partially constant Dirac delta

I am trying to integrate the product of a function and a partially constant delta function over a sphere of constant radius $r$. The integral is of the form $\int^{2\pi}_0 \int^{\pi}_0 ...
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2answers
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$\iiint_Ez(x^2+y^2+z^2)^{-3/2}dV$ where $E=\{(x,y,z):x^2+y^2+z^2\leq16, z\geq2\}$

Convert to spherical coordinates and evaluate: $$\iiint_Ez(x^2+y^2+z^2)^{-3/2}dV$$ where $E$ is the region satisfying the following inequalities: $x^2+y^2+z^2\leq16$, $z\geq2$. When I drew out the ...
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2answers
86 views

Using spherical coordinates to find volume of a region

Use spherical coordinates to find the volume of the region lying above $z = \sqrt{3x^2+3y^2}$ and within the $x^2+y^2+z^2=2az$, $a>0$. So far I know that the first graph is a cone and the second ...
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49 views

How to compute the following Jacobian

I need to show that the Jacobian of the n-dimensional spherical coordinates is $$\displaystyle r^{n-1}\sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots\sin\phi_{n-2}$$ then I have computed the Jacobian matrix, ...
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106 views

How to compute triple integral in spherical coordinates

I need to compute: $\displaystyle\int \int \int z dxdydz$ over the domain: $\left\{x^2+y^2+z^2\leqslant 16,z\geqslant 0\right\}$ Im trying to use spherical coords as: \begin{equation} ...
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1answer
411 views

Transforming a matrix from cartesian to spherical coordinates

Consider a variable matrix $$\left[\begin{array}{ccc}a_{11}(x,y,z) \quad a_{12}(x,y,z) \quad a_{13}(x,y,z)\\ a_{21}(x,y,z) \quad a_{22}(x,y,z) \quad a_{23}(x,y,z)\\ ...
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2answers
197 views

Maximum Gravity Around a Unit Sphere

Most simulations that involve planetary gravity use Newton's law of universal gravitation and treat planets like point-masses. This is very accurate at large distances, and fairly accurate all the ...
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1answer
2k 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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1answer
146 views

Laplacian on Sphere of Function Only Depending on Angle Between Points

Consider a function $f:S^2 \to \mathbb{R}$ , with $S^2$ the unit $2$-sphere in $\mathbb{R}^3$. Let's say that $f$ depends only on the polar angle $\theta$ from the north pole (e.g., $f(r,\theta,\phi) ...
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integrating a vector over a sphere

I have the following triple integral in spherical coordinates $(r,\theta,\phi)$: $$\int_0^{2\pi}\int_0^\pi\int_0^RCr^3\hat\theta\cdot r^2dr\sin{\theta}d\theta d\phi$$ How do I handle the ...
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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 ...
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2answers
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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 ...
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Volume of $n$-dimensional spherical orthant in upper diagonal halfspace

Consider an $n$-dimensional Euclidean Space. Consider orthants in that space. Each orthant occupies $\frac{1}{2^n}$ of the volume of an $n$-dimensional unit sphere. Let's call that a spherical ...
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41 views

Integrating a sphere by discs vs shells (spherical coordinates)

I am getting very confused about the following. Let's say I want to find the volume of a sphere. I can start with a circle having circumference $2\pi R\cos\theta$. I can multiply by $R d\theta$ and ...
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111 views

Is there a spherical coordinates system for vectors of complex numbers?

Suppose I have a scalar field $f(\vec{x})$, where $\vec{x}\in\mathbb{R}_3$, and I wish to average $f$ over a sphere $|\vec{x}|=R$: $\displaystyle\langle f\rangle_{R} = \frac{\int_{S} f(\vec{x})\, ...
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On the equidistant distribution of $n$ points on a sphere $S^2$ by algorithm and their “validity” measures by statistical methods

I have found an algorithm for distributing $n$ points $P_0, P_1, ..., P_n$ (approximately) equidstantly on a sphere where $$\varphi_i = \pi(\phi - 1)i \qquad \theta_i= \mathrm {asin} (2i/n - 1), ...
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1answer
103 views

Formula for Determinant of Vectors given in spherical coordinates

In 2D, one has an easy formula for the determinant of two vectors given in spherical coordinates, i.e. $\begin{vmatrix} \cos(\phi_1) &\cos(\phi_2)\\ \sin(\phi_1) &\sin(\phi_2)\end{vmatrix} ...
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89 views

Computing volume element in spherical coordinates

Suppose $y = (r, \theta^1, \theta^2)$ are spherical coordinates in $(\mathbb{R}^3,g)$. What is the $d\text{vol}$ in these coordinates? I solved it but I don't know if it's right. My solution: We ...
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Converting a polar integral to spherical

$$\int_0^{2\pi} \int_0^{\sqrt{2}}\int_r^{\sqrt{4-r^2}}\mathrm{d}z \, r \, \mathrm{d}r \, \mathrm{d}\theta$$ So in spherical this would become: $$\int_0^{2\pi} \int_0^{\pi/4}\int_0^2 \rho^2\sin\phi \, ...
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1answer
236 views

Integration in n-spherical coordinates

I'd like to compute the following integral: $$I = \int_{\mathbb{R}^n} {\rm d}^n x \; \frac{e^{i \vec x \cdot \vec k}}{\vec x^2}$$ My first step is to use generalized spherical coordinates and then I ...
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2answers
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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 ...
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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 ...
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663 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, ...
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2answers
280 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 ...
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1answer
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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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2answers
113 views

How to show $\DeclareMathOperator{curl}{curl}\curl\curl(e_r) = 0$

I want to figure out how to calculate $\text{curl}(e_r$). Where $e_r$ is a base vector for the Spherical co-ordinate system. Taking $e_r = (\sin\theta \cos\phi)i+(\sin\theta ...
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3answers
66 views

2-Sphere surface coordinate dimension

Ordinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. If we use ...
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Equation for calculating azimuth between two points

Does anybody know an equation or approximation for calculating the azimuth as a function of latitudes and longitudes of both the points. For example I have Princeton, NJ is at 40.3571° N, 74.6702° ...
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2answers
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Discretize a circle on a sphere with a given center and radius

I would like to draw a discretized circle on the surface of a sphere (the Earth in that case). The input of the algorithm would be the center of the circle (expressed as longitude and latitude), the ...
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4answers
2k 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.
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1answer
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Integration of function $\sqrt{1 - \sum_{i}x_{i}^{a_{i}}}$ by using spherical coordinates in D dimensions

Let's have integral $$ I = \int \limits_{0}^{\text{T.P.}}dx_{1}...dx_{D}\sqrt{1 - \sum_{i}x_{i}^{a_{i}}}, \quad \text{T.P.}: \quad 1 - \sum_{i}x_{i}^{a_{i}} = 0 $$ By rewriting it in coordinates ...