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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calculate volume in spherical coordinates
In calculate volume enclosed within the cone of $z^2=4(x^2+y^2)$ and inside sphere $x^2+y^2+(z-2)^2=4z-3$ in spherical coordinates.
what's limit of $\rho$?
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0answers
39 views
Distance on the surface of a sphere
Given a sphere which radius is r.
There are two red points on the sphere. Given the location of the two points in spherical coordinate system.
If the surface distance between a point and a red point ...
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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 ...
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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 ...
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. ...
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1answer
44 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} ...
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1answer
28 views
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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1answer
37 views
Triple Integral Spherical Coordinates
So I have to compute the triple integral of this:
$\int\int\int \frac{1}{1+x^2+y^2+z^2}$ and it says the equation of the sphere is $ x^2 + y^2 + z^2 = z$ which is just an elongated sphere running ...
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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 ...
0
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1answer
40 views
Volume of a 3D sphere of radius $R$ using Riemannian metric in stereographic coordinates
The question is pretty much in the title. We were also given the hint that it could be useful to use spherical coordinates when calculating the integral (the actual answer is not required, just its ...
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 ...
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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}$.
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1answer
62 views
Finding a third coordinate on a sphere that is equidistant from two known coordinates
Here is my problem that I'm having some trouble with: I have the coordinates (latitude and longitude) of two points on Earth. I have no problem finding the great circle distance between the two ...
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2answers
161 views
Find volume of a cone $z=k\sqrt{x^2+y^2}$ bounded by $z=h$ using spherical coordinates
We were given this exercise in class to take home but I am a bit confused with it. If anyone could help I would appreciate it.
Let $C$ be a conical solid bounded above by $z=h$ and below by the cone ...
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0answers
28 views
Help me obtain the pressure equation for a gas in a spherical coordinates
Assume a sphere filled with $n$ gas particles moving freely.
if we draw a circle with an area of $dA$ on the inside wall of the sphere, how many particles with velocity $v$ and angles $\theta$ and ...
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0answers
37 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 ...
3
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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 ...
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1answer
47 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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0answers
59 views
two points on a unit sphere
Consider the two vectors to the points on the unit sphere,
$${\bf v}_i=(\sin\theta_i\cos\varphi_i,\sin\theta_i\sin\varphi_i,\cos\theta_i)$$
with $i=1,2$. Use the dot product to get the angle $\psi$ ...
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vote
2answers
99 views
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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vote
2answers
54 views
Simple approach for geo distance
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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0answers
34 views
Turn any shape to circle
I'm looking at trying to calculate and re position 3d vectors to align in a new position to form a circle.
I've achieved this already however only when all points are evenly distributed in the same ...
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0answers
169 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) = ...
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 ...
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1answer
72 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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0answers
115 views
Generating evenly distributed points on a sphere
How could I write an algorithm to generate n points distributed 'evenly' on a sphere? I already wrote an algorithm to generate points distributed uniformly on the surface (here), but by 'evenly' ...
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1answer
57 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 ...
0
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0answers
66 views
Centre of a spherical triangle
Suppose I have a triangle defined by 3 unit vectors {$v_1, v_2, v_3$} in a 3 dimensional complex inner product space.
What would be the centre of such a triangle? I guess it should be something like ...
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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 ...
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vote
0answers
178 views
Rotating co-ordinates in 3D
Suppose I have 3 axes, $x$, $y$, and $z$ such that $x$ is horizontal, $y$ is vertical, and $z$ goes in/out of the computer screen where $+$ve values stick out and $-$ve values are sunken in.
Suppose ...
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1answer
159 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} $$
where the ...
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1answer
241 views
How to find the 3D coordinates on a celestial sphere's surface?
With celestial I don't mean a normal sphere, but I mean one that uses the altitude and an azimuth angle system. This is what I mean for example:
(the star in the image represents an example of a ...
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2answers
152 views
Identifying a surface $\rho^2\cos(2\phi)-1=0$
I need convert this spherical expression, to a rectangular form (specific surface). $$\rho^2\cos(2\phi)-1=0$$
Thanks for a while.
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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 + ...
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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 ...
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1answer
96 views
Spherical coordinates to cartesian coordinates.
I want to find out the distance between the centers of $2$ circles.
Say, circle $1$ $(\theta,\phi)$
circle $2$ $(\theta,\phi)$
The radius of this circle is found using $d\tan(\theta)$
where $d$ is ...
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vote
1answer
85 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 radius of ...
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1answer
51 views
Spherical Coordinates
Let Q be the region above by the plane $8z=4-x-y$ and below by the cone $64z^2=x^2+y^2$. How would I setup the triple integral to find volume of Q, using spherical coordinates? I just need help with ...
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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 ...
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vote
1answer
110 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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1answer
74 views
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$$
and a line through the center of the sphere with equation :
$x=a+\alpha$ , ...
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vote
1answer
476 views
Divergence Theorem to evaluate integral where S is a sphere
Use the Divergence Theorem to evaluate
$$\iint_S x^2y^2+y^2z^2+z^2x^2 dS$$
where $S$ is the surface of the sphere $x^2+y^2+z^2=1$.
So the divergence of F is $2y^2x+2z^2y+2x^2z$. Then I tried to ...
3
votes
1answer
85 views
Integration on a sphere
I have an integral at hand which has the form of
$$I = \int_{u\in \mathbb{S}^2} f(\mathbf{u}\cdot \mathbf{s}_1) f(\mathbf{u}\cdot \mathbf{s}_2) d\mathbf{u}$$
where $\mathbb{S}^2$ is the unit sphere ...
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1answer
641 views
Great arc distance between two points on a unit sphere
Suppose I have two points on a unit sphere whose spherical coordinates are $(\theta_1, \varphi_1)$ and $(\theta_2, \varphi_2)$. What is the great arc distance between these two points?
I found ...
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1answer
70 views
How many coordinates are necessary to determine a sphere?
Do determine a circle, you would need at least three coordinates. How many are necessary to determine a sphere?
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votes
2answers
371 views
Projection of a 3D spherical distribution function in to a 2D cartesian plane
Consider a 3D spherical Gaussian distribution function that depends on radius only,
$$f(r) = \frac{1}{N} e^{-(\frac{r-R_\mu}{\sigma})^2}$$
where $R_\mu$ is the radial offset of the distribution and ...
0
votes
1answer
50 views
Coordinates transformation and falling body trajectory
As it's well known, assuming the earth fixed and non rotating, the trajectory of a falling body with initial speed $v_0 = \{v_{0x},v_{0y},{v_{0z}}\}$ is contained in a plane. Assuming an observer in ...
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1answer
809 views
Equation for making a circle in 3D space
I have a 3D space with axis $(x, y ,z)$ and I can make a circle in the $xy$-plane.
To make a circle in the xy-plane I currently use spherical coordinates $(r, \theta, \phi)$ where $r = 1$, $\theta = ...
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1answer
196 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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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 & ...
