Continuous, Slope-Preserving Maps of Simple Closed Curves
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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).
1976 ◽
Vol 19
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pp. 373-374
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2018 ◽
Vol 10
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pp. 323-354
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1992 ◽
Vol 34
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pp. 314-317
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1960 ◽
Vol 24
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pp. 163-172
2013 ◽
Vol 6
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pp. 4-8
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1960 ◽
Vol 12
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pp. 209-230
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