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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3
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1answer
63 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 ...
0
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1answer
23 views

$\iiint_W\sqrt{x^2+z^2}\,\mathrm{d}V$ Where $W$ is the solid delimited by $y=4$ and $y=\sqrt{x^2+z^2}$

$\iiint_W\sqrt{x^2+z^2}\,\mathrm{d}V$, $W$ is limited by the plane $y=4$ and the paraboloid $y=\sqrt{x^2+z^2}$. I'm trying to solve with spherical coordinates, however I got stuck in the following ...
1
vote
1answer
18 views

What is the easiest way to find the radius and center of the circle of intersection between two spheres?

If given two spheres $S_1$ and $S_2$, of radius $r_1$ and $r_2$, centered at 3-space points $P_1$ and $P_2$, respectively. What is the easiest way to find the radius and center of the circle of ...
0
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1answer
27 views

Determine coordinate of a point on unit sphere

Let $S$ be unit sphere in $\mathbb R^3$ center at $O(0,0,0)$. Let $A=(x_1,y_1,z_1),B = (x_2,y_2,z_2)$ be two points lying on the sphere $S$. Let $M$ be center of $AB$ which lies on the geodesics ...
3
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1answer
58 views

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 ...
1
vote
1answer
33 views

Proving $v$ is harmonic

Let $u$ be a harmonic function in $\mathbb{R}^3$ and let $a > 0$. Show that the function $v$ defined in spherical coordinates by $v(r,\theta,\psi )=\frac{a}{r}u(\frac{a^2}{r},\theta,\psi)$ ...
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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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0answers
15 views

How to integrate a distribution in spherical coordinates over a circle?

I have an angular distribution $\frac{s \sigma}{d\Omega} = \frac{d\sigma}{d(cos\theta)d\phi}$. How can I calculate it over a circle which lies on the plane $X = dist$, has radius $r$ and its centre is ...
-2
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2answers
246 views

Find the center of a circle on a sphere given 2 points and its radius [closed]

How can I find the set of the centers of the circles on sphere that pass through 2 given points and have pretedermined radius, using spherical coordinates? Assume that the radius of sphere is $R$. ...
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0answers
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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0answers
20 views

Find a point on extended great circle with given distance

Given 2 points on earth with longitude and latitude coordinate A(lat1, lon1), B(lat2, lon2), and a distance d. Find coordinate (in longitude and latitude) of 2 points C, and D on extended of great ...
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1answer
15 views

Particularly Problematic Spherical Polar Problem

The question is as follows: Using spherical polar coordinates, find the volume of the solid specified by R $\leq$ 3 and $0 \leq \theta \leq \frac{\pi}{3} $. I have two big questions about this ...
3
votes
1answer
262 views

why is the laplacian of 1/r equal zero outside the origin?

In spherical coordinates, the laplacian can be written as: $\nabla^2 = \frac{1}{r}\frac{\partial^2}{\partial r^2}r + \frac{1}{r^2 \sin \theta}\frac{\partial}{\partial \theta}(\sin \theta ...
1
vote
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 ...
0
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1answer
33 views

Triple Integral with Spherical Polar Coordinates Problem

I am having trouble understanding the whole concept of spherical polar coordinates, so any help is appreciated! The full question is: Let D be the region bounded by the surface of the hemisphere $z = ...
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0answers
15 views

Is the $r$-axis in spherical coordinates the same as the $z$-axis?

If Cartesian coordinates have an $x$-axis, $y$-axis, and $z$-axis, do spherical coordinates have an $r$-axis, a $\theta$-axis, and a $\phi$-axis? Since the Cartesian $z$-axis is just the set: ...
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0answers
15 views

How to intergate the cross section over the surface of a detector?

My beam moves along the $X$ axis. I know the cross section $\frac{d \sigma}{d \Omega}$. My rectangular detector is perpendicular to the $XY$ plane and its surface is perpendicular to the line ...
3
votes
1answer
79 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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1answer
36 views

Dual Spaces vs Dual Bases

I'm trying to wrap my head around differentiable manifolds and tensors. I partially worked through a question which asked me to use the metric tensor and the line element in spherical polar ...
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0answers
33 views

Marsden Triple Integration, Spherical Coordinates

This comes from Marsden Vector Calculus book. I think I've done it correctly, but I'm not very confident in my work. The integral is $$\int_0^3\int_0^\sqrt{9-x^2}\int_0^\sqrt{9-x^2-y^2} ...
2
votes
1answer
59 views

How to integrate a part of a sphere limited by a rectangle?

I have a function which depends on the solid angle $\frac{df}{d \Omega} = \frac{df}{d\phi d\theta} $. I want to integrate it over a part of the sphere limited by a rectangle. How should I set the ...
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0answers
21 views

Coordinate system for evenly-distributed points on a sphere

Suppose I have a sphere with a set of evenly-distributed points on its surface: What is the coordinate system to use to represent these points, where each adjacent point is a unit step? Spherical ...
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0answers
14 views

Relationships between positions on a sphere in a global coordinate system

Imagine I have a camera which is observing an object. The camera can move around the abject, by traversing a virtual sphere, where the radius of the sphere is the distance between the object centroid ...
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0answers
57 views

First person camera rotation (conversion between spherical to cartesian coordinates)

A similar question has been asked before on Stack Overflow but no satisfying answer. Basically I'm following this tutorial, and I can't exactly figure out how the guy deduced these two equations to ...
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0answers
92 views

Triple integral in spherical coordinates: Finding $\phi$ limits?

The question asks to convert to spherical coordinates then evaluate. So for this question, I manage to get the bounds of theta and row right, but I got the bounds of phi wrong. According to the ...
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0answers
32 views

Is it possible to calculate the lat/lon of a point knowing the extent coordinates?

I would like to know where it's possible to determine the latitude and longitude corresponding to a point inside a cartesian (planar?) representation of a geographical map, knowing the x/y coordinates ...
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0answers
29 views

Finding moment of intertia using spherical coordinates

I have taken a picture of the problem with my work. The answer I am getting is incorrect. I think I am doing something wrong after the following. Any help is appreciated `
0
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1answer
33 views

Parameterising of scalarproduct of surface gradient with vector field

Given is this operation $\left((\mathbb{1}-\hat{r}\hat{r})\cdot \nabla\right) \cdot v(r) = \left(\nabla-\hat{r} (\hat{r} \cdot \nabla)\right) \cdot v(r)$, where $r=(x,y,z) \in \mathbb{R}^3$, ...
0
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1answer
64 views

Find volume of region lying above $z=0$, below $z=4-x^2-y^2$ and inside extruded disc $x^2+y^2=2^2$

I am working on the following homework problem: Find the volume of the region that lies above the plane $z=0$, below the surface $z=4-x^2-y^2$ and inside the extruded disc $x^2+y^2=2^2$. I think ...
0
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2answers
181 views

Evaluate the integral by changing to spherical coordinates.

$$\int_{0}^{6} \int_0^{\sqrt{36-x^2}} \int_{\sqrt{x^2+y^2}}^\sqrt{72-x^2-y^2} xy~ dzdydx $$ I tried converting it and I ended up with ...
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0answers
19 views

Spherical to Cartesian coordinate ellipsoid overlap

I have two geographic coordinates; latitude and longitude , separated by few meters. I need to draw an ellipsoid of same major and minor axes ,centered around the geographic co-ordinates I used ...
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0answers
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 ...
3
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2answers
42 views

$\iiint_Ez(x^2+y^2+z^2)^{-3/2}dV$ 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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0answers
10 views

Scale factors and metric in cylindrical and spherical coordinates - isotropy of space [duplicate]

In cylindrical (polar) coordinates, the scale factors are $$h_r=1$$ $$h_{\theta}=r$$ $$h_z=1$$ Would it be correct to say that $h_i$ do not depend on $\theta$ because space is isotropic (has the same ...
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0answers
13 views

Shortest distance (in km) between a lat/long and a line between two lat/longs?

I know that Haversine can be used to calculate the absolute distance between two lat/long, but is there any implementation of the shortest distance (in km) between a lat/long and the line that is ...
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1answer
30 views

dot-product spherical

I want to calculate dotproduct of $ e_r *e_\phi $ they are unit vectors in spherical. where my spherical coordinates is $(r,\phi,\theta)$ My attempt is first to convert them to cartesian: which ...
2
votes
2answers
51 views

cylindrical coordinates - base vector integral

Hello I have a problem where $\hat{p}$ is a basevector in a cylindrical system $$\int_{0}^{\pi/4}\hat{p} d\theta$$ I know that; $\hat{p} = \hat{x}\cos\theta + \hat{y}\sin \theta$ $$ ...
2
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1answer
38 views

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 ...
1
vote
1answer
38 views

What is the equivalent of <x,y,z> in spherical coordinates?

I know this is a very newbie question, but what is the equivalent of $\langle x,y,z \rangle$ in spherical coordinates? I'd think it would be $\langle r, \theta, \phi \rangle$ but the divergences ...
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2answers
64 views

Calculating the percent of area “covered” by a vector pointing on a sphere

The question is inspired by rotational dynamics and how much of the sky could a camera "cover" when it rotates in a specific way. Let's say that we have solved the equations for rotational motion of a ...
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2answers
29 views

Converting from cylindrical to spherical coordinates?

I am supposed to convert the point $(100, -\dfrac{\pi}{6}, 50)$ from cylindrical to spherical. The $\rho$ and $\theta$ are easy ( $\sqrt{100^2+50^2}, \dfrac{\pi}{6}$ respectively) but the $\phi$ is ...
0
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1answer
30 views

Convert a density function $\rho(r)$ of sphere to ellipsoid.

Note: This is not a homework problem. As the title somewhat eludes to, I have a density function for a sphere as a function of radius $\rho(r)$. I would like to then flatten the sphere slightly into ...
2
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0answers
62 views

Integration over n-sphere

I am trying to integrate squares of sum of coordinates in n-dimensional sphere with radius bounded by r. $f(x,y)=x^2+y^2$. In spherical coordinates when $N=2,\;y=a\cos\theta$ and $x=a\sin\theta$ So ...
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0answers
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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2answers
81 views

Create a rectangle with coordinates (latitude and longitude)

I have two points on a map, I want to create a rectangle where the two points are the line that intersect the rectangle. I understand that there is no true rectangle on a sphere, but the areas I am ...
1
vote
1answer
58 views

Formula to convert Cartesian coordinates to spherical coordinates? [closed]

I have this formula: x, y, z = cos(vertical)*sin(horizontal), sin(vertical), cos(vertical)*cos(horizontal) Which maps a spherical coordinates (horizontal and ...
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0answers
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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0answers
24 views

Given how long it took the sound to reach two points, find how long it takes to reach the third point

I am having some issues trying to solve this puzzle. I am given 3 gps locations as starting points on a map. I am given the time it takes for the sound of a item being placed on the group in number of ...
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0answers
26 views

Spherical coordinate system remain the same if the origin is changed and each point make the same translation?

Now I have a spherical coordinate system whose origin is located at (a,b,c)[cartesian], and I have another point whose location is (r, theta, tho) in this spherical system, and P's cartesian ...
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2answers
82 views

Computing a double integral over a surface S, where S is the unit sphere,

$$ \int \int_S (x^2+y^2)d\sigma$$ Where S is the unit sphere centered at (0,0,0), and $\sigma$ is surface area. I arrived at the correct answer of $\large \frac{8\pi}{3}$, but I took an (educated?) ...