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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Changing from rectangular coordinates to spherical coordinates (integration)

I am taking calculus 3 and I have problems understanding how to change from rectangular coordinates to spherial ones (integration). For example, I have this problem: Find the volume of the solid $T$ ...
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188 views

Law sines in Spherical Triangle $\rightarrow$ Law sines in plane triangle

Could any one tell me how to estimate or get law of sines in Spherical Triangle to The Law of Sines in Plane Triangle? i.e $\frac{\sin a}{\sin A}=\frac{sin b}{\sin B}=\frac{\sin c}{\sin C}$ to ...
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2answers
275 views

Calculating spherical distance between two geo-locations

I wanted to show my nephew(16) a simple approach to calculate the distance between two geo-locations. The mathematical knowledge of a 16-year old boy is limited to simple geometrical shapes like ...
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1answer
334 views

Distance measurement between latitude/longiture pairs.

I need to calculate the distance between two lat/lng coordinate pairs. In addition, If given an initial lat/lng coordinate, angle of travel, and distance, I need to calculate the resulting lat/lng ...
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415 views

Integration on the unit sphere

I have an integral on the unit sphere as follows. $$I(\mathbf{s}_1, \mathbf{s}_2) = \int_{\mathbb{S}^2} f(\mathbf{x} \cdot \mathbf{s}_1)f(\mathbf{x}\cdot\mathbf{s}_2)d\mathbf{x} $$ where the ...
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453 views

Converting between spherical coordinate systems

Say I have the spherical coordinates of some locations, specifically their longitude ($0$ to 360) and latitude (latitude = $0$ at equator, $90$ at north pole, $-90$ at south pole) on a sphere with a ...
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327 views

Analytically derive n-spherical coordinates conversions from cartesian coordinates

I'm finding it difficult to find any non-geometrical derivation of coordinate conversions from cartisan to spherical. I can understand the derivations geometrically, because I can visualize the ...
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1k views

Divergence Theorem to evaluate integral where S is a sphere

Use the Divergence Theorem to evaluate $$\iint_S x^2y^2+y^2z^2+z^2x^2 dS$$ where $S$ is the surface of the sphere $x^2+y^2+z^2=1$. So the divergence of F is $2y^2x+2z^2y+2x^2z$. Then I tried to ...
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974 views

Monte Carlo simulation on sphere: unbiased random steps

Im doing a Metropolis Monte Carlo simulation with particles on a sphere and have a question concerning the random movement in a given time step. I understand that to obtain a uniform distribution of ...
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1answer
903 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 ...
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227 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: ...
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78 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 ...
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159 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 ...
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19 views

shperical coordinates limits

how to write the limits for the integration in spherical coordinates to get the volume of 1- the solid inside the cylinder $$r= \sin(\theta)$$ bounded from above by $$z=\sqrt{1-x^2-y^2}$$ and ...
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23 views

Getting topological objects from the “cube” of $T^3$

One can imagine $T^3$ much like he can imagine $T^2$: as a flat box with opposite faces identified. One may put coordinates on $T^3$, each of which would logically range from $0$ to $2\pi$. To get ...
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1answer
40 views

Is there a solution to the equation $tan({\phi})=\frac{0}{0}$

I've been reading about conversion from Cartesian ($x,y,z$) to Spherical (r, $\theta$, $\phi$) coordinates. The formula to find the value of ${\phi}$ is given as: $\tan({\phi})=\frac{y}{x}$ My ...
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49 views

Estimating the distance between two coordinates but without using Euclidean distance

Bill opens up "Café Finder" on his phone, and it tells him that it will take him 10 minutes to get to his nearest Starbucks to grab a triple-shot frapa-crapa-flat-white, so he decides to walk. 20 ...
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41 views

Integration in spherical coordinates

I needed to solve the following integral on one of my exercise sheets, which seemed not too difficult: $ \phi(\vec{r}) = \dfrac{1}{4\pi\epsilon_0} \int\limits_0^{\infty} dr' \int\limits_0^{\pi} ...
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16 views

How to setup/evaluate a triple integral to show an interesting result in physics?

I know this isn't the physics forum, but the task i'm struggling with is purely mathematical. My task is as follows; Let $A$ be a sphere centered at origin with radius $R$ and assume $a \geq R$. ...
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31 views

Marginals/conditionals of a normalized Gaussian vector

It is well - known that if $x=(x_1,...,x_n)^T\sim{N(0, \sigma^2I)}$, then its normalized version is uniformly distributed on the unit $n-1$ - sphere: $$ ...
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216 views

Triple integral to find the mass of the intersection between two spheres

I've got two unit spheres, one is centered at $(0,0,0)$ and the other at $(0,0,1)$, the intersection of these two spheres is my region $R$. I would like to find the integral: $$\iiint\limits_R ...
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20 views

Vectors and dot product in spherical coordinate system [duplicate]

Let $\vec{v_1}$ and $\vec{v_2}$ given: $\overrightarrow{V_1} = r_1\hat{u_r} + \theta_1\hat{u_\theta} + \phi_1\hat{u_\phi} \\ \overrightarrow{V_2} = r_2\hat{u_r} + \theta_2\hat{u_\theta} + ...
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angle between hrizontal and a line connecting the center of an oblate ellipse to a point in space

I would like to know how I can calculate the angle $\alpha$ in an oblate ellipse similarly to the sphere.
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rotation of spherical surface in spherical coordinates

I need to plot a spherical surface in computer (like the surface of a lens). I know the normal vector (as an example, say $\ n=(1,2,3) $) of this surface and it originates from the centre of the ...
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25 views

Spherical Triangle — Angle from one Point without using North

I'm designing a tool for students and right now im working with coordinates. I have the plan to shift the north pole in my code to Munich and I would like to do this by working with a spherical ...
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1answer
27 views

Find critical points using spherical coordinate

again! Let $E:=\left\{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 \le 4 \right\}$ and $f:E \to \mathbb{R}$ the function define by $$f(x,y,z):= \frac{z}{1+x^2+y^2+z^2}.$$ How can I determine the ...
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39 views

Locate a point on sphere with equal distance

Given 3 points A (lat1, lon1), B(lat2, lon2), O(lat3,lon3) on earth with geometric location longitude and latitude and a distance d, where O is middle point of A and B. Let GCD denote the great ...
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1answer
25 views

Volume of Region using Spherical Polar Coordinates

Find the volume of the region D bounded by the hemisphere $y=\sqrt{4-x^2-z^2}$ and the planes $y=x\ $,$\ y= \sqrt3x$ by using polar coordinates. Working: I have calculated $x^2+y^2+z^2=4$ so am I ...
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48 views

How to get parallels of tilted Equator?

I have a Great Circle on Earth, which is not an Equator nor Meridian, and it's not parallel to these. I have four geographical coordinate pairs for it, separated by 90 degrees, so I can use these in ...
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58 views

Coordinates of an n-sphere [duplicate]

I'm a little embarrassed to ask this question because it should be easy but it's stumped me for over a week now. The answer will determine how I write some code, so it matters. According to Wikipedia ...
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30 views

Numerical evaluation of the Laplace operator near the singularity points in spherical coordinates

If one had to evaluate $\Delta Y_l^m$ numerically everywhere on the unit sphere, including the singularity points $\theta = 0,\pi$, how would they do it? Let's say $Y_l^m$ is a spherical harmonic. I'm ...
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3answers
61 views

Polar Co-ordinate proofs

The expression for acceleration in spherical polars is $$ \ddot{\mathbf r} =( \ddot r -r\dot\theta^2-r\dot\phi^2\sin^2\theta) \mathbf e_r + (r\ddot\theta+2\dot r ...
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30 views

Integrating over particular grids to obtain Spherical Harmonic coefficients

Theoretically the spherical harmonic expansion coefficients of a function $f$ should be calculated via a continuous integration: $$F_{lm} = \int_{0}^{2\pi}\int_{0}^{\pi} ...
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63 views

Spherical Parametrization of a Cholesky Decomposition

[Background: Trying to build up my math base knowledge (self-taught) to follow an explanation on page 291 of this document, dealing with spherical parametrization to estimate unknown ...
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1answer
160 views

PDE Heat Equation with Variable Coefficient {Second ODE Variable Coefficient}

Another PDE question: If I have a non constant coefficients in my heat equation (PDE), how do I solve it? For example we have: $\frac {\partial T}{\partial t} =\frac {\partial ^2 T}{\partial r^2} + ...
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21 views

Cover the entire sphere with a moving cone

Assume you have a cone with half-angle theta in a 3D cartesian space, with the vertex of the cone in the origin, and that you want to rotate the cone along a curve so that it covers the entire sphere. ...
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82 views

Problem with center of mass in polar coordinates

When we calculate center of mass using rectangular coordinates, we find the average values in each coordinate. Obviously we can't do this very same thing in polar coordinates: if we integrated a ...
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135 views

distance in n-dimensional space

According to answer of this question : Distance between 2 points in 3D space (in spherical polar coordinates) The distance between 2 points in 3 dimensional space is : $$ ...
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1answer
255 views

Center of mass of a trick sphere-cone intersection

B is the solid region occupying the space situated inside the sphere of radius R centered at the origin and above the cone of equation $z = \sqrt{x^2 + y^2}$. The B density is proportional to the ...
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How do you represent a vector that points to an arbitrary point on a sphere of radius R?

Is it just $$v = R \hat{r} + \theta\hat{\theta} + \psi \hat{\psi}$$ That doesn't seem right as the unit is not correct.
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389 views

Set up integral in spherical coordinates outside cylinder but inside sphere

I have the equation of a cylinder and the equation of a sphere given: Cylinder: $x^2+y^2=4$ Sphere: $x^2+y^2+z^2=25$ I'm asked to set this up in cylindrical and spherical coordinates. Cylindrical ...
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103 views

3D rotational matrix between two spherical co-ordinate systems.

So I have a classical mechanics problem where I have worked out the azimuthal and altitude angle for a vector, I then want to apply rotational matrices so that the vector is realigned with the z axis ...
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57 views

Convert from Spherical to Cylindrical Coordinates

The following integral is given in Spherical Coordinates, which procedure should I follow to express it in Cylindrical Coordinates? $$\int_{0}^\pi \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} ...
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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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integrate over quadrant of sphere

It is known that $\int_{x\in S}\exp(\kappa\mu^Tx)dx$ where S is the surface of the unit sphere is $\frac{(2\pi)^{p/2} I_{p/2-1}(\kappa) }{\kappa^{p/2-1}}$ where $p$ is the number of dimensions and $I$ ...
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2answers
362 views

Spherical harmonic expansion of a sphere

Seeing as one can expand any function on the sphere in terms of the spherical harmonics, I was thinking it should be possible to express the function for a sphere itself in terms of them. I have ...
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147 views

Convert geodetic coordinates to cartesian coordinates

I am working on some simulation software that will represent a number of entities in a defined geographic area in the world. The part of the software that I am currently working on is to implement ...
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Why did my teacher integrate \varphi in the opposite direction while solving a triple integral for volume in spherical coordinates?

In Calculus III class today we learned how to evaluate triple integrals in spherical coordinates. One of the example questions we worked on was to use a triple integral to find the volume of a shape. ...
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156 views

Line-of-Sight Angle on Sphere

I'm trying to calculate the angle (in degrees) between two latitude/longitude pairs, but with a twist. Most calculations I see use the Great Circle / bearing method, but this does not seem correct ...
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angle difference in hyper-spherical coordinate

The question is kind of intuitive: Consider a points $p$ in $\mathbb{R}^d$ ($d$-dimensional Euclidean space). $$p=\{x_1, x_2, \ldots, x_d\}$$ We can always transform it into spherical coordinate. ...