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Continuous, Slope-Preserving Maps of Simple Closed Curves

Published online by Cambridge University Press:  20 November 2018

Tibor Bisztriczky  and
Ivan Rival
Tibor Bisztriczky
Affiliation:
University of Calgary, Calgary, Alberta
Ivan Rival
Affiliation:
University of Calgary, Calgary, Alberta
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How many of the continuous maps of a simple closed curve to itself are slope-preserving? For the unit circle S1 with centre (0, 0), a continuous map σ of S1 to S1 is slope-preserving if and only if σ is the identity map [σ(x, y) = (x, y)] or σ is the antipodal map [σ(x, y) = (–x, –y)]. Besides the identity map, more general simple closed curves can also possess an “antipodal” map (cf. Figure 1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1102 - 1113
Copyright
Copyright © Canadian Mathematical Society 1980

References

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