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FLIPPABLE SURFACE

A flippable surface is a surface globally invariant under the action of a half-turn (or axial symmetry).

A surface with Cartesian equation  can be identified as flippable if their exists a half-turn r of  such that .
With the + sign, the half-turn does not swap the two faces of the surface; examples:
    - all the surfaces of revolution
    - the ellipsoid, the centered quadrics, and more generally all the surfaces with equation  that are invariant under action of the three half-turns around the axes.
    - the cross-cap, and more generally all the surfaces with equation  that are invariant under the action of the half-turn around Oz.
 

With the - sign, the half-turn swaps the two faces of the surface; taking the axis of the half-turn to be equal to the line  we get a general implicit equation of these surfaces:  with ; examples:

- the plane z = 0
- the hyperbolic paraboloid
video 1
- Plücker's conoid
video 2
- the symmetric parabolic Dupin cyclide
video 3
- the surface 
video 4
- Costa's algebraic surface
video 5
- the surface 
video 6

- the Enneper minimal surfaces, and Costa's minimal surface.

REMARK: all the surfaces of the above box have an equation of the type ; their isometry group is composed of the identity, the half-turn around Oz that does not swap the faces, the two half-turns around , two reflections, and two rotorotations of order 4, this group is isomorphic to that of the isometries of the square.

See more generally the surfaces with rotational symmetry.
 
 
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© Robert FERRÉOL  2017