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

Yoshihiko Tazawa
Organization: Tokyo Denki University
Department: Natural Sciences
Journal / Anthology

Innovation in Mathematics: Proceedings of the Second International Mathematica Symposium
Year: 1997
Page range: 459-466

In this article we will show how to use Mathematica in dealing with curves and surfaces in the three dimensional unit sphere S3 embedded in the four dimensional Euclidian space E4. Since S3 is the Lie group of unit quaternions and at the same time it is a space of constant curvature, the analogy of the theory of curves in E3 holds. We calculate curvature and torsion of curves in S3 by Mathematica. The Gauss map v of a surface in E4 is decomposed into the two maps v+ and v_. If the surface is contained in S3, we can define another Gauss map vs. 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.

*Mathematics > Calculus and Analysis > Differential Geometry
*Mathematics > Geometry > Surfaces