Math

Klein bottle and the projective plane

Explore a Klein-bottle immersion and a Roman-surface image of the projective plane. Follow a local normal and inspect the identifications behind non-orientability.

By Mykhailo Makarov

Actual application image

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Follow the normal arrow around a non-orientable loop. The Klein bottle is a self-intersecting immersion; the Roman surface represents antipodal points of a sphere and has singularities.

Equations used in the module

Equations
Klein identification: (u+2π,v) ~ (u,−v)
RP² = S²/(n ~ −n)
A continuous local normal returns reversed on an orientation-reversing loop
Klein in R³ self-intersects; Roman image has singular points

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