Folding a Circle into A Cone - Parameterization Mathematica Notebook

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"My project is to take the circle's folding up into a cone using the
coordinate system in: https://zenodo.org/record/7710313 from the
difference between the circumferences of two circles, and have a
mathematica notebook that animates the transformation using the
coordinate systems specifically."

Cone nets are possible for the vertex angle of π/2,, 3π/2,2π and so on. The
transformation from a point (x, y) in the grid plane to the point (x′, y′, z′) on
the cone is given by