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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1answer
95 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
111 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 ...
0
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1answer
73 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
103 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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2answers
429 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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2answers
176 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
71 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 ...
2
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0answers
484 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) = ...
2
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2answers
185 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 ...
0
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1answer
237 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
329 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
193 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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0answers
155 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 ...
1
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1answer
205 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 ...
1
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0answers
484 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
282 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 ...
3
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1answer
866 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
476 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.
2
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1answer
153 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 + ...
4
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1answer
324 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 ...
-1
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1answer
171 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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1answer
229 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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1answer
76 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 ...
3
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3answers
8k 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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1answer
158 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
92 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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1answer
964 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
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1answer
133 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 ...
2
votes
2answers
5k 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
122 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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2answers
814 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 ...
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1answer
69 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 ...
1
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1answer
2k 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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2answers
516 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 ...
2
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2answers
664 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 & ...
0
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1answer
80 views

Getting coordinate between two coordinates knowing the distance and latitude

That is my wall: http://imageshack.us/f/266/wall2z.png/ I know the coordinates of the lower points (left and right). (X1,Y1,Z) and (X2,Y2,Z) where X es the latitude, Y longitude and Z the altitude. ...
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0answers
65 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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0answers
155 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 ...
4
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4answers
2k views

Why is the differential solid angle have a $\sin\theta$ term in integration in spherical coordinates?

When you integrate in spherical coordinates, the differential element isn't just $ d\theta d\phi $. No. It's $\sin\theta d\theta d\phi$, where $\theta$ is the inclination angle and $\phi$ is the ...
2
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2answers
154 views

Are there derivations of equations of non-degenerate real quadric surfaces

Take the ellipsoid for example $$(x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1$$ in the x-y plane you have an ellipse described by $$(x^2/a^2)+(y^2/b^2)=1$$ (suppose z=constant) in the y-z plane you have an ellipse ...
2
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1answer
2k views

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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0answers
68 views

Shooting projectiles with GPS locations

I need a formula that given a GPS location a cannon ball will be shot from there, and I need to find the other GPS location that the cannon ball will fall. I can get direction, considering a circle ...
1
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1answer
251 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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1answer
2k views

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 ...
1
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1answer
461 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 ...
1
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1answer
184 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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1answer
831 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 ...
3
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1answer
388 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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4answers
863 views

What is the geodesic between a point and a line (geodesic between two points) on an oblate spheroid?

I found a similiar question that also asks for the distance from a point to a line but works on a sphere. Now I'm trying to figure out the length of the geodesic line ...
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1answer
222 views

Relatively simple geometry with latitude/longitude coordinates

I'm trying to calculate a user's deviation from a route defined as a list of latitude/longitude points that make up the route. My proposed solution is to try and calculate the distance between the ...