Rhyno_DaGreat
Make X the Dividend and Y the Divisor.
Thus:
X/Y
Now... it's usually read two different ways: "The amount of Y that can go into X" or "The amount of X per Y".
Ah. You mixed that up a bit. The first interpretations should either be "the number of times Y can go into X"
or "the amount (portion) of X that is equal to Y."
Rhyno_DaGreat
Does anyone understand how the spherical coordinate system is derived? If anyone could explain it to me, that'd be great!
Well, the easiest thing to do is to transform from Cartesian (x,y,z) coordinates. The radial coordinate r is defined as the distance from the origin, which, by the distance formula (which is really just a repeated Pythagorean theorem), is
r = sqrt[x²+y²+z²].
The zenith angle φ is defined as the angle the line from the origin to (x,y,z) makes with the positive z-axis. From the image to the left, where z is the distance to the xy plane (hence forming a right triangle), you can see that the sine of the complement of φ is (opposite/hypotenuse) = z/r. But of course the sine of the complement if φ is simply cos φ. Therefore:
φ = arccos(z/r).
[This may be slightly more obvious if you imagine a line from (0,0,z) to (x,y,z), although that's not drawn in the diagram, in which case, cos φ = z/r directly.]
Also in the diagram, observe that the distance from the origin to the projection to the xy plane, i.e., (x,y,0), is then r' = r sin φ. The azimuthal angle θ is defined that the counterclockwise angle from the positive x axis to (x,y,0) in the xy plane. But this is then just a repeat of polar coordinates:
θ = arctan(y/x)
and also
x = r' cos θ, y = r' sin θ,
which is equivalent to
x = r cos θ sin φ, y = r sin θ sin φ, z = r cos φ.
Note: It is a common convention to reverse θ and φ compared to the diagram (e.g., among physicists), so be clear which convention your source is using. See also
here.
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