Experiments in Differential Geometry with Mathematica
Tokyo Denki University
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The classical theory of curves and surfaces was established long ago. However, there
has always been difficulty in applying the general theory to individual curves or surfaces
because calculating integrals or solving differential equations explicitly is not possible
in general. So the examples appearing in this field have been confined to a small group of
calculable objects. Mathematica has made it possible to deal with a more
generic group of objects. In this talk I will introduce some experimental visualizations
that I have produced with Mathematica and used in my differential geometry class.
They help students to understand basic notions of differential geometry without
difficulty. Two examples are: 1) to generate a random closed differentiable plane curve
and observe the change of its curvature, Gauss map, and total absolute curvature under the
homotopy between the unit circle, and 2) to approximate a space curve with pieces of
circles joined together twisted at each joining point according to the torsion.