Editing Geodesic (section)

== Ribbon test ==
{{Multiple image
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| caption1 = The Ribbon Test 
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| caption2 = The curved line drawn using the ribbon test is a straight line on a flat surface. This is because a cone can be made into a 2-d circular sector.
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A ribbon "test" is a way of finding a geodesic on a physical surface.<ref>{{Cite AV media |url=https://www.youtube.com/watch?v=Xc4xYacTu-E |title=Which Way Is Down? |date=2017-11-02 |last=Vsauce |access-date=2025-03-26 |via=YouTube}}</ref> The idea is to fit a bit of paper around a straight line (a ribbon) onto a curved surface as closely as possible without stretching or squishing the ribbon (without changing its internal geometry).

For example, when a ribbon is wound as a ring around a cone, the ribbon would not lie on the cone's surface but stick out, so that circle is not a geodesic on the cone. If the ribbon is adjusted so that all its parts touch the cone's surface, it would give an approximation to a geodesic.

Mathematically the ribbon test can be formulated as finding a mapping <math>f: N(\ell) \to S</math> of a [[neighborhood_(mathematics)|neighborhood]] <math>N</math> of a line <math>\ell</math> in a plane into a surface <math>S</math> so that the mapping <math>f</math> "doesn't change the distances around <math>\ell</math> by much"; that is, at the distance <math>\varepsilon</math> from <math>l</math> we have <math>g_N-f^*(g_S)=O(\varepsilon^2)</math> where <math>g_N</math> and <math>g_S</math> are [[metric_tensor|metrics]] on <math>N</math> and <math>S</math>.