Geodesic
Continuous, Slope-Preserving Maps of Simple Closed Curves
Canadian journal of mathematics, Tome 32 (1980) no. 5, pp. 1102-1113

Voir la notice de l'article provenant de la source Cambridge University Press

How many of the continuous maps of a simple closed curve to itself are slope-preserving? For the unit circle S 1 with centre (0, 0), a continuous map σ of S 1 to S 1 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). 
Bisztriczky, Tibor; Rival, Ivan. Continuous, Slope-Preserving Maps of Simple Closed Curves. Canadian journal of mathematics, Tome 32 (1980) no. 5, pp. 1102-1113. doi: 10.4153/CJM-1980-084-4
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