An Interesting Stereographic Projection

This is an interesting stereographic projection that you get when you rotate 90-degree Steiner circles about their axis of symmetry.

Animated GIF; you may need to click on it for running:

This is an example LaTeX file for a particular value of \animationparameter, that you can run yourself here in the browser:

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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)}%
}
\pgfmathdeclarefunction{toruscenter}{1}{%
    % #1  - \animationparameter
    \pgfmathparse{1/cos(#1)}%
}
\pgfmathdeclarefunction{torusradius}{1}{%
    % #1  - \animationparameter
    \pgfmathparse{-sin(#1)/cos(#1)}%
}
\pgfmathdeclarefunction{projectionx}{3}{%
    % #1  - \animationparameter
    % #2  - \longitude
    % #2  - \latitude
    \pgfmathparse{
        torusradius(#1) * 
        sphereX(#2,#3) +
        toruscenter(#1) * 
        cos(#2)
    }%
}
\pgfmathdeclarefunction{projectiony}{3}{%
    % #1  - \animationparameter
    % #2  - \longitude
    % #2  - \latitude
    \pgfmathparse{
        torusradius(#1) * 
 
 
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