All possible immersions of a torus in 3D Euclidean space can be grouped into four regular homotopy classes. All possible immersions within one such class can be transfigured into one another through continuous smooth transformations that will put no tears, creases, or other regions of infinite curvature into the surface. This report introduces four simple, easy-to-understand representatives for these four homotopy classes and describes several transformations that convert a more complex immersion of some torus into one of these representatives. Among them are transformations that turn a torus inside out and others that will rotate its surface parameterization by 90 degrees.
Title
Torus Immersions and Transformations
Published
2011-07-20
Full Collection Name
Electrical Engineering & Computer Sciences Technical Reports
Other Identifiers
EECS-2011-83
Type
Text
Extent
27 p
Archive
The Engineering Library
Usage Statement
Researchers may make free and open use of the UC Berkeley Library’s digitized public domain materials. However, some materials in our online collections may be protected by U.S. copyright law (Title 17, U.S.C.). Use or reproduction of materials protected by copyright beyond that allowed by fair use (Title 17, U.S.C. § 107) requires permission from the copyright owners. The use or reproduction of some materials may also be restricted by terms of University of California gift or purchase agreements, privacy and publicity rights, or trademark law. Responsibility for determining rights status and permissibility of any use or reproduction rests exclusively with the researcher. To learn more or make inquiries, please see our permissions policies (https://www.lib.berkeley.edu/about/permissions-policies).
