Steiner Circles

If you take the stereographic projection of a mesh sphere, you obtain Steiner circles.

Interestingly, the map of any inscribed circle of the sphere under stereographic projection is a perfect circle on the plane.

Unfortunately, due to the low sampling size, the smooth option can be a bit botched sometimes.

The code is parameterized so that the user can specify Euler angle rotations of the sphere, and see the resulting stereographic projection on the plane.

If a line goes near the north pole, we omit it by setting the opacity to zero.

There are other workarounds, but they require using a conditional in a for loop, and I wanted to use the smooth option.

For example, this animation shows the sphere undergoing a rotation which is based on all three angles changing differently:

Try changing the values of \zrotationa, \yrotation and \zrotationb for yourself!

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\documentclass[
    tikz
    ,border = 3.14mm
]{standalone}
\usepackage{tikz-3dplot}
\pgfmathdeclarefunction{sphereX}{2}{%
    % #1  - longitude
    % #2  - latitude
    \pgfmathparse{cos(#2)*cos(#1)}%
}
\pgfmathdeclarefunction{sphereY}{2}{%
    \pgfmathparse{cos(#2)*sin(#1)}%
}
\pgfmathdeclarefunction{sphereZ}{2}{%
    \pgfmathparse{sin(#2)}%
}
% https://tex.stackexchange.com/a/736767/319072
\pgfmathdeclarefunction{tdplottransformrotmainX}{6}{%
    % #1  - Z rotation
    % #2  - Y rotation
    % #3  - Z rotation
    % #4  - x value
    % #5  - y value
    % #6  - z value
    \pgfmathparse{
        (
            (
                sin(#1) * cos(#2) * cos(#3) + 
                cos(#1) * sin(#3)
            ) * #4 +
            (
                -sin(#1) * cos(#2) * sin(#3) + 
                cos(#1) * cos(#3)
            ) * #5 +
            sin(#1) * sin(#2) * #6
        ) * -1
    }%
}
\pgfmathdeclarefunction{tdplottransformrotmainY}{6}{%
    \pgfmathparse{
        (
 
 
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