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Cone Geodesics

Geodesics on a cone are easily found using the fact that the surface is isometric to the plane. The left image shows a line specified by two parameters,  (distance from the origin) and  (angle between the normal vector and the horizontal axis). The right part shows a cone together with the geodesic that represents an isometric image of the given line.

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DETAILS

Assuming that the plane is parametrized using polar coordinates as and the cone as , the isometry between the two surfaces is given by , .
In rectangular coordinates, the line in the plane is parametrized as , . These coordinates have to be transformed to polar coordinates before applying the isometry.
Making the surface semitransparent gives a better overview of the geodesic behavior, but decreases the visualization performance.
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