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May 12th, 2014, 04:16 AM  #1 
Newbie Joined: May 2014 From: Australia Posts: 1 Thanks: 0  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 
May 12th, 2014, 09:36 AM  #2 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108 
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 xy (equator) plane (azimuth angle), and they use $\theta$ to measure how much a vector is elevated above the xy 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 zaxis (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 800002:2009 "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 radiusvectors. 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. 
May 12th, 2014, 01:13 PM  #3  
Global Moderator Joined: May 2007 Posts: 6,542 Thanks: 592  Quote:
 
May 12th, 2014, 01:17 PM  #4 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  

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coordinates, geometry, involving, latitude, longitude, question, spherical 
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