17 April 2014
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Paper Information

Curves and surfaces in the three dimensional sphere placed in the space of quaternions

Author(s): Yoshihiko Tazawa

Abstract:
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...

Pages: 8
Size: 455 kb
Paper DOI: 10.2495/IMS970591

 

 

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Innovation in Mathematics

Innovation in Mathematics

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