Conversion of coordinates to screen coordinates of part of LabVIEW

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  This is a convention that I use and which seems to be acceptance quite wide (I'm not sure that's the convention of Wolfram, however).

  Let X, Y and Z is a right coordinate system.  That P is a point whose X, Y, Z coordinates we know.  Define a system of spherical polar coordinates (SSC) as follows:

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  • Polar angle (phi) - the angle between the axis Z (or 'pole') and the line from the origin to P.  By convention, including phi between 0 and pi.

  Under this system of coordinates, the following code (very similar to Altenbach, which uses a slightly different coordinate system) cards X, Y, Z in Rho, Theta, Phi:

  

  We will check certain quantities of 'known '.  Start with the x-axis of the unit, [1, 0, 0].  This clearly has length, Rho = 1.    The X axis lies in the XY, so the angle between the latter and the Z axis is 90 °, Phi should be pi/2.  Finally, the projection of the X axis in the XY plane is located along the X axis, so the angle between the two axes of X is 0, so Theta should be 0.  The first Re / Im polar will give 1, Theta Yes 0 = 0.  The second will give 1, pi/2, and Yes 1 = Rho and Phi = pi/2.

  How about unit Z, [0, 0, 1]?  Again, the length, Rho, is 1.  Technically, since the z plans on the origin, the Azimuthal angle cannot be properly defined (the X coordinates and are 0), so we'll do a loophole 'math' and just say that it is zero.  The polar angle is easy - the angle between the Z axis and the pole, the Z axis is 0, so Phi = 0.  Now "do the math".  Put [0, 0] in the first Re / Im polar gives [0, 0], so theta = 0.  The second d / Im to polar takes [1, 0] [1, 0], Rho Yes = 1 and Phi = 0.

  It is easy to show (in particular by the coding of this place and put in test of numbers) that negative quantities are working properly.  Rho is always non-negative, Phi is always in the range 0... PI and theta of pi-pi, with the sign depends on the sign of Y (as it should).

  Bob 'just do the math' Schor

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