Coordinate transformation: Spherical to Cartesian I'd like to calculate the x, y, z distance to an object given its spherical coordinates theta and phi and it's altitude above the earth. My origin is on the surface of the earth so I don't know the rho coordinate of the object.  
I think I can do that using the x, y, z formulas listed here but I am not sure how to calculate the distance that the object is from me (rho).  How should I approach this?  I'd like to assume the earth is a simple sphere for this exercise.
I took a look at this MATLAB page and it mentions a "broadside" formula but I am not sure how my angles line up with the angles that it mentions.
Example:
Altitude = 100
Theta = 30deg
Phi = 30deg
I've thought about trying to make a triangle with one side being radius of the earth, the second side being radius of earth + altitude and the third side being rho.  I am not sure how to calculate the angles of that triangle.
 A: Steps for a "naive" way.


*

*Given any two points $\mathbf r$ and $\mathbf r'$, the distance between them is $d(\mathbf r, \mathbf r') = \sqrt{(x-x')^2 + (y-y')^2 + (z-z')^2}$.

*Determine the cartesian components $x,y,z$ and $x',y',z'$ for both points from the spherical coordinates you currently have.

*Plug the expressions from the last step into the formula in the first step.


To deal with the altitude, notice that your position is just $\mathbf r = (R + h)\hat{\boldsymbol \rho}$ where $\hat{\boldsymbol \rho}$ is the radially outward-point unit vector $R$ is the radius of the Earth, and $h$ is your height above the surface, and there are standard expressions for that unit vector that you can look up.
A: I think I figured it out.
You can calculate rho by considering a triangle:
The central angle would be 180 - theta and the two sides would have length of radius of the earth and radius of the earth + altitude.  The third side can be figured out with the law of sines.
Once you have rho, you can calculate x, y, z based on how the the spherical coordinate system is specified (directions of +x, +y, +z).  
