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May 12th, 2014, 04:16 AM   #1
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Geometry question involving spherical coordinates and longitude and latitude.

Hi, I'm almost completely stuck on this question: http://i.imgur.com/SpUJYCm.png [1]
I realize that p must be equal to the earth's radius if the point is on the surface of the sphere. My strategy, for now, is to first find the values for theta and psi for both Melbourne and London. From there I was thinking converting them into Cartesian coordinates to easily set up a plane equation... Not sure if this will work though.
So far I haven't gotten past the first stage. I'm having trouble converting the longitude latitude coordinates into theta and psi. I'm getting really confused by the definitions.
So, how do you convert (37.81 S, 144.96 E) and (51.51 N, 0.13 W) into psi and theta? If you could also throw in some advice on whether i'm heading in the right direction from there, that would be awesome as well.
Thanks very much
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May 12th, 2014, 09:36 AM   #2
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Please read some text that explains spherical coordinates and their conversion into Cartesian coordinates, such as your textbook, lecture notes or Wikipedia. Note that, unfortunately, there is a considerable disagreement on notation. First, some people use $\phi$ for the angle in the x-y (equator) plane (azimuth angle), and they use $\theta$ to measure how much a vector is elevated above the x-y plane. Others (in particular, calculus textbook authors in the US) do it the other way around. (Aren't inches, yards and miles enough?) Second, some people of the first tradition use $\theta$ for the angle between the vector and the z-axis (inclination, or polar angle), while others use it to denote the angle between the vector and the equator plane (elevation angle). Obviously, the sum of inclination and elevation is $\pi/2$. In the geographical coordinate system, latitude is elevation.

According to the international standard ISO 80000-2:2009 "Mathematical signs and symbols to be used in the natural sciences and technology", $\phi$ (more precisely, $\varphi$) denotes the azimuth angle (on the equator plane) and $\theta$ (in fact, $\vartheta$) denotes the inclination.

In the following picture, I denoted radius by $\rho$, azimuth angle by $\phi$ and latitude by $\alpha$. Depending on your convention, $\theta$ may be equal to $\alpha$ or to $\pi/2-\alpha$.



Taking this into account, read the Wikipedia article, especially the conversion formulas in the "Cartesian coordinates" section. Again, note that inclination = $\pi/2$ - latitude.

Once you know how to find Cartesian coordinates, us the dot product to find the cosine of the angle between the two radius-vectors. I recommend expressing it in general through $\rho$, $\phi_1$, $\phi_2$, $\theta_1$ and $\theta_2$ and then substituting the values of angles. Once you know the angle between the vectors, it's easy to find the distance.
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May 12th, 2014, 01:13 PM   #3
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Quote:
Originally Posted by theguyoo View Post
Hi, I'm almost completely stuck on this question: http://i.imgur.com/SpUJYCm.png [1]
I realize that p must be equal to the earth's radius if the point is on the surface of the sphere. My strategy, for now, is to first find the values for theta and psi for both Melbourne and London. From there I was thinking converting them into Cartesian coordinates to easily set up a plane equation... Not sure if this will work though.
So far I haven't gotten past the first stage. I'm having trouble converting the longitude latitude coordinates into theta and psi. I'm getting really confused by the definitions.
So, how do you convert (37.81 S, 144.96 E) and (51.51 N, 0.13 W) into psi and theta? If you could also throw in some advice on whether i'm heading in the right direction from there, that would be awesome as well.
Thanks very much
You don't need to convert to cartesian coordinates. Latitude and longitude are angles. Latitude origin is the equator (domain -90°,90°), longitude origin is Greenwich (domain -180°,180°). Switching to standard spherical coordinates is simply shifting the origin for latitude and the domain for longitude.
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May 12th, 2014, 01:17 PM   #4
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Originally Posted by mathman View Post
You don't need to convert to cartesian coordinates.
The problem asks to calculate the surface distance between the two points and not just their spherical coordinates.
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