WIT Press

Curves And Surfaces In The Three Dimensional Sphere Placed In The Space Of Quaternions


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Yoshihiko Tazawa


In this article we will show how to use Mathematica in dealing with curves and surfaces in the three dimensional unit sphere S^ embedded in the four dimensional Euclidian space E\ Since S* is the Lie group of unit quater- nions and at the same time it is a space of constant curvature, the analogy of the theory of curves in E^ holds. We calculate curvature and torsion of curves in S^ by Mathematica. The Gauss map v of a surface in E^ is decomposed into the two maps v+ and z/_. If the surface is contained in 5^, we can define another Gauss map z/g. We use Mathematica to visualize the shapes of the images of these Gauss maps. Finally, the meaning of these images becomes clear through the notion of the slant surface. 1 Curves The space E* is re