Converting cartesian coordinates into latitude and longitude coordinates

spherical coordinatesspherical-geometry

I'm trying to convert a position vector on a unit sphere into the latitude and longitude coordinates but I'm not sure how to do it. I know that the formula for converting the latitude and longitude coordinates into cartesian coordinate is: $$V_n=\begin{pmatrix} R\cos(\lambda_n) \cos(\phi_n) \\ R\sin(\lambda_n) \cos(\phi_n) \\ R\sin(\phi_n) \end{pmatrix}$$ where $\lambda$ represents the longitude and $\phi$ represents the latitude.

Could I get some advice on how to convert cartesian coordinates into latitude and longitude?

Best Answer

Take $V_n/R$. The last element, $z/R=\sin\phi_n$, can be used to get $\phi_n$. You can use $\arcsin$ function, since $\phi_n$ is between $-\pi/2$ and $\pi/2$ in radians. You might need to convert to degrees, so the answer will be between $-90^\circ$ and $90^\circ$. For $\lambda_n$, take the ratio of $y$ and $z$ components, and you can see that you get $\tan\lambda_n$. To get the answer in a $360^\circ$ range, use the $\mathrm{arctan2}$ function

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[Math] Spherical distance between two points in terms of latitude and longitude

Indeed both the formulae are identical. This can be shown in the following manner.

In the spherical polar coordinate system $\theta$ polar angle is the angle between the zenith direction and a line segment OP. But $\theta$ is nothing but the co latitude. Since we do know that latitude

$\theta$ = 90 - colatitude and also $\cos(90 - \theta)$ = $\sin\theta$. Substituting this into the original identity reveals that the identities are equal.

Clamp Spherical coordinates to latitude/longitude

If you want efficiency, using $\text{arccos}(\cos \phi)$ is like driving 5 miles south to the ring-freeway, following it around to the north side of town, and driving another 5 miles south to visit the second house up the block from where you started. Similarly for $\text{atan2}$. You would be hard-pressed to come up with a more in-efficient solution.

Seriously - computers directly add, subtract, multiply, and divide. Anything else takes multiple steps to calculate - usually many such steps. It used to be that even multiplying and dividing required multiple steps, but then chip technology improved until the whole logic could be performed in one step. And yes, on the most advanced chips, some functions - square rooting in particular - are now incorporated into a single operation. But even if the trig functions were all incorporated, why would you bother when the problem has a much more simple computation?

A simple algorithm takes care of it:

ToLatitudeLongitude(phi, theta)    
   phi = mod(phi, 2*pi)
   theta = mod(theta, 2*pi)
   if (phi > pi) 
      phi = 2*pi - phi
      if (theta < pi) 
         theta = theta + pi
      else 
         theta = theta - pi

   return (phi, theta)

Best Answer

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[Math] Spherical distance between two points in terms of latitude and longitude

Clamp Spherical coordinates to latitude/longitude

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