Difference between revisions of "Ring torus"
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(Created page with "==Definition== The '''ring torus''' is a form of embedding of the torus in threedimensional Euclidean space. This surface type is ''not'' unique up to isometry or even up to si...") 
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Revision as of 06:45, 12 August 2011
Definition
The ring torus is a form of embedding of the torus in threedimensional Euclidean space. This surface type is not unique up to isometry or even up to similarity transformations, but rather, depends on two parameters for a description up to isometry and on one parameter for a description up to similarity transformations.
Implicit and parametric descriptions
Degree of generality  Implicit description  What the parameters mean  Parametric description  What the additional parameters mean  Comment 

Arbitrary  Fill this in later  
Up to rigid motions (rotations, translations, reflections)  is the radius of the central circle (spine) of the ring torus, and is the tube radius of the ring torus. This describes the ring torus where the axis of revolution is the axis.  is an angle giving local polar coordinates for the point any fixed location of the circle being rotated. is the angle giving polar coordinates for the center of the circle, on the spine circle.  
Up to similarity transformations  We could rescale the above to normalize either one of and to 1, but we cannot normalize both simultaneously. 