Antipodally symmetric nongeodesic spherical curve

For sufficiently small nonzero

ε

γ (θ) = \frac{( c o s θ , s i n θ , ε s i n 3 θ )}{1 + ε ^{2} s i n ^{2} 3 θ}

(90)

is an embedded antipodally invariant curve on the round sphere. It is not a great circle and hence not a geodesic, while the antipodal isometry exchanges its two complementary discs.

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Geodesic curvature
Differential geometry
Geometry and topology
Area of mathematics
Mathematics
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Solution