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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16 views

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. ...
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1answer
100 views

How to describe the region inside a sphere and below a cone in cylindrical and spherical coordinates?

If E is the region of space located inside the sphere $x^2 + y^2 + z^2 = 4$ and below the cone $z = \sqrt{3x^2 + 3y^2}$ How may I describe E in cylindrical and spherical coordinates? And how may I ...
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62 views

Looking for algorithm for spherical point in polygon that works across meridian and anti-meridian

I need to process millions of latitude/longitude points every day to see if they are located within a defined lat/lon bounded polygon. The polygon may be rectangular, or it may be some irregular 3.. ...
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2answers
521 views

Equation of a straight line in spherical coordinates

I'm trying to prove the angle sum formula for a triangle on the surface of a sphere. In order to do this I wanted to create a general triangle on the sphere, with one vertex at $\theta = 0$ and one ...
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32 views

Parametrization of a bounded solid.

So, I have a solid bounded by $z=\sqrt{x^2+y^2}, z=\sqrt{1-x^2-y^2}, z=2$ I had to parametrize it using spherical coordinates so I used $$\begin{cases} x(\rho, \theta, ...
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74 views

Calculate a point on a geodesic line on an ellipsoid

I have a problem which i don't understand how to achieve. Maybe someone could sheed some light on it. Have a look at this picture: What I try to achieve is to determine the point D on the geodesic ...
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1answer
75 views

Implicit partial derivative of a spherical cap

Consider a spherical cap, for which the base radius is $a$ and the height is $h$. Then, the surface area and volume is (these equations can be found on Wolfram Mathworld) $A(a,h) = \pi(a^2 +h^2)$, ...
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90 views

How to find Latitudes and Longitudes of projections of the vertices of a rectangular plane below earth's surface?

I want to find out the latitudes and longitudes of projections of the vertices of a rectangular plane inside the earth's surface. I know dimensions of rectangle, angles of orientation and latitude and ...
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32 views

Projecting non-central Gaussian on a sphere?

Suppose I have an $n$-dimensional Gaussian random variable with mean $\mu\in \mathbb{R}^n$ and covariance matrix $\sigma^2I_n$, where $I_n$ is the identity matrix. If I condition on the distribution ...
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56 views

volume in spherical coordinates

I am trying to find the region bounded by the sphere $p = 2\cos\psi$ and hemisphere $p=1$, $z\geq 0$. Not quiet sure not to do this problem, so please help.
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1answer
54 views

Distortion in spherical coordinates

I'm trying to realized 3d models of stones. My idea was to create a 2D random angular distribution with opportune correlation, namely $R(\theta,\varphi)=rand(\theta,\varphi,c_l)$ where ...
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49 views

Proof of $\vec(r) \times \nabla$ in spherical coordinates

My professor claims, that $\vec{r} \times \vec{\nabla} = \vec{e}_{\varphi} \frac{\partial}{\partial \vartheta} - \vec{e}_{\vartheta} \frac{1}{r\sin \vartheta} \frac{\partial}{\partial \varphi}$ in ...
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1answer
80 views

Numerical solution needed for the quadratic equation (spheres)

There are three point coordinates $A(x_1,y_1,z_1),B(x_2,y_2,z_2),C(x_3,y_3,z_3)$, where $z_1=z_2=z_3$ Now, $(x-x_1)^2+(y-y_1)^2+(z-z_1)^2=d_1^2$ this equation can be written for my system as ...
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2answers
111 views

Computing distance from line to point in geodetic environment

Supposing to be in a cartesian plan and that I have the following point: $$A(x_{1},y_{1}), B(x_{2},y_{2}), C(x_{3},y_{3}), D(x_{4},y_{4})$$ $$P(x_{0},y_{0})$$ Now immagine two lines, the fist one ...
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0answers
73 views

Integrating Over Truncated Sphere

Consider the volume bounded by the sphere of radius R (with center $r_0$ = R/2 z) and the plane z = 0. Evaluate $\int dS$ over the surface of the truncated sphere. I'm not sure how to integrate over ...
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243 views

Conversion from Spherical Coordinates to Cartesian Coordinates aligned along arbitrary polar axis

I have spherical coordinates $w = (\theta, \phi)$ such that $\theta$ is the angle between $w$ and the polar axis (let's assume $z$ is up). Assuming $w$ is a unit vector, the conversion to cartesian ...
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1answer
85 views

How to derive curl in spherical coordinates

This is one of those questions where I know I am making a dumb mistake someplace and I am trying to check where it is. $$ \begin{vmatrix} \frac{\bf\hat r}{r^2\sin\theta} & ...
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1answer
149 views

Angled Spherical Sector - formulas?

I am glad to be here. First off, please excuse my almost saddening lack of knowledge. Math (in general) isn't my strong point. I might ask some really basic questions, you have been warned :) The ...
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75 views

Joint PDF for spherical region

A sphere has a coordinate system (r, $\theta$, $\phi$) with the origin at the center of the sphere. What is the joint PDF of the r and $\phi$ coordinates, $f_{r,\phi}(r,\phi)$, for a randomly ...
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1answer
230 views

Directions in spherical coordinates

Say I have a system with standard spherical coordinates. There's a man on that sphere and he's standing on the equator facing east. He chooses a random angle $0°-360°$ and turns that much in the clock ...
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352 views

Distance of two points in spherical coordinates

Today I was trying to solve a physics exercise and ran into some mathematical problems. Consider two concentric spheres $K_{1,2}$ with radii $R_{1,2}$. I wanna solve the following integral: $$E_{ij} ...
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490 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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206 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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634 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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74 views

Knowing coordinates of a point having two coordinates and the distance.

I have the two geographic coordinates of the lower corners of a wall. So, for example, i want to know what is the coordinate that is for example 15cm on the right of the lower corner left. Is that ...
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176 views

Cross-section of a circle with a three-dimensional Gaussian

Suppose I have a three-dimensional Gaussian with mean $\bar{\mu}$, volume $A$ and covariance matrix $\Sigma$ $$G(X)=\frac{A}{\sqrt{(2\pi)^{3}\det(\Sigma)}}e^{-\frac{1}{2}(X-\mu)^{T}\cdot ...
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1answer
272 views

dotting gradient in spherical coordinates with displacement vector

The gradient in spherical coordinates is given by: $\nabla f = \left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial ...
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194 views

Relative spherical coordinates of 2 points.

I have 2 points in space, defined by their spherical coordinates. I'd like to know the spherical coordinates of the second point in a reference system centered on the first point (I know the unit ...
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0answers
162 views

What is the name for the axis that “inclination angle” is measured from in spherical coordinates?

What is the name for the axis that "inclination angle" is measured from in spherical coordinates? I keep trying to call it "polar axis", and the other axis from which the azimuth is measured, the ...
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3answers
75 views

Triple Integral in Spherical Coordinates.

$\newcommand{\de}{\operatorname{d}}$A little stuck on this one. $$\iiint_V ye^{-(x^2+y^2+z^2)^2}\,{\rm d} V$$ Use Spherical Coordinates to evaluate where V is the solid that lies between y=0 and the ...
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1answer
68 views

Granted I have NE and SW coordinates for a rectangle, how do I get the center point?

I've got the NE and SW coordinates/points for a minimum bounding rectangle. How do I calculate the center point of this rectangle? At first thought, I could calculate this using simple division. ...
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2answers
56 views

How do I calculate opposite / most distanced coordinates on the earth?

If i get this coordinates: City Coordinates: 43°52′0″N 18°25′0″E φ Latitude °N, λ Longitude °E (of Map center): (43.8562586, 18.413076300000057) How do i determine most distant / opposite ...
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1answer
67 views

Find the volume inside

Find the volume inside the torus $\rho=\sin\phi$. First of all how can $\rho=\sin\phi$ represent a torus? I can't even visualise that. All Ideas are welcome, this looks like a 'food for thought ...
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1answer
143 views

Finding the limits of a triple integral.

Evaluate: $$\iiint_V (x^2+y^2)\:\mathrm{d}x\:\mathrm{d}y\:\mathrm{d}z,$$ where $V$ is the region in the positive octant bounded by the sphere $\|\vec{r}\|=a$. I am unsure how to get the limits of ...
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2answers
40 views

Divergence of vector in spherical coordinates

How should I calculate the divergence for $$\vec{V}=\frac {\vec{r}}{r^2}$$ Is it possible to convert it from spherical coordinates to cartesian?
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2answers
699 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
1k views

Converting from Spherical to Rectangular

I need to convert $\rho \sin\phi=2\cos\theta$ in to rectangular form. Attempt: I tried using those nice properties : $$x=\rho\sin\phi\cos\theta \\y=\rho\sin\phi\sin\theta\\z=\rho\cos\phi$$ and ...
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3answers
60 views

The geometric meaning of certain mappings written in cylindrical or spherical coordinates

What is the geometric meaning of the following mappings, that are written in cylindrical coordinates? The mappings are: $$(r, \theta, z) \rightarrow(r, \theta , -z) \\ (r, \theta , z) \rightarrow (r, ...
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1answer
301 views

Drawing ellipse as google.maps.Polygon with 8 points

In a web page using Google Maps JavaScript API v3 (including Geometry library) I currently draw an ellipse as a "diamond" with 4 corner points by the following JavaScript code: ...
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2answers
32 views

Derivation for the integrating term in line integrals and volume integrals in spherical coordinates

Can anyone refer me to, or respond with, the derivation for the integrating term in line integrals $dl=dr\hat{r}+rd\theta\hat{\theta}+r\sin\theta\ d\phi\hat{\phi}$ and volume integrals ...
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2answers
47 views

How can I solve these two tough integrals?

\begin{equation*} J_{1} = \int_{0}^{\sqrt{{\pi}/{6}}} \int_{y}^{\sqrt{{\pi}/{6}}} \cos{(x^2)}\,dx\,dy \end{equation*} \begin{equation*} J_{2} = \int\int_{E}\int z e^{(x^2+y^2)} + xe^{x^8}\,dV, ...
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1answer
27 views

Conversion to Spherical Coordinates

http://www.physics.usu.edu/Wheeler/QuantumMechanics/QMOrbitalAngularMomentum.pdf Unfortunately, this is the first time I've come across a conversion to spherical coordinates and I'm pretty lost. The ...
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2answers
501 views

How to move a camera in “Flight Simulator” style (Roll, Pitch, Yaw) using a joystick

I am working on a video game where the camera movement (what the viewer sees) is controlled by a joystick. I want the camera movement to act like a flight simulator meaning the following: When the ...
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2answers
697 views

How to calculate the angle between two vectors, defined by 3 points on the earth?

I want to develop a formula to calculate the angle between two vectors. The vectors will be OX and OY (from point O to X , and Y), where the points are defined by their latitude and longitude values. ...
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1answer
128 views

Exterior Product $d\Phi_1\wedge d\Phi_2$ and spherical coordinates

One short question: If $\Phi\colon\mathbb{R}^3\to\mathbb{R}^3$, defined by $$ \begin{pmatrix}r\\\vartheta\\\phi\end{pmatrix}\mapsto\begin{pmatrix}r\sin \vartheta\cos \phi\\r\sin \vartheta ...
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1answer
496 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
447 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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1answer
86 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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1answer
11 views

How do I get vectors orthogonal to the one generated by the spherical coordinate formula?

Given a formula: F : ℝ → ℝ → ℝ3 F(θ,φ) = (cos(φ)*sin(θ), sin(φ)*sin(θ), cos(θ)) what are the formulas: ...
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3answers
30 views

Cylindrical - Spherical coordinates

We are given a point in cylindrical coordinates $(r, \theta , z)$ and we want to write it into spherical coordinates $(\rho , \theta , \phi)$. To do that do we have to write them first into ...