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.