Dual Transformation Between $$S{widehat{O}}(3)$$ S O ^ ( 3 ) and $$S{widehat{O}}(2,1)$$ S O ^ ( 2 , 1 ) and Its Geometric Applications

摘要

This paper defines a dual transformation between dual-special orthogonal matrices $$S{widehat{O}}(3)$$ S O ^ ( 3 ) and dual-special semi-orthogonal matrices $$S{widehat{O}}(2,1)$$ S O ^ ( 2 , 1 ) . This transformation makes clear the relations how orthogonal axes look like in 3-dimensional dual space $$D^{3}$$ D 3 and 3-dimensional dual-Minkowski space $$D_{1}^{3}$$ D 1 3 . Also, the similarities between $$E^{3}$$ E 3 and $$E_{1}^{3}$$ E 1 3 is given by drawn some pictures of rotational surfaces using Mapple13.