next surface  previous surface  2D curves  3D curves  surfaces  fractals  polyhedra 
FLIPPABLE SURFACE
A flippable surface is a surface globally invariant under the action of a halfturn (or axial symmetry).
A surface with Cartesian equation
can be identified as flippable if their exists a halfturn r of
such that .
With the + sign, the halfturn 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 halfturns around the axes.
 the crosscap,
and more generally all the surfaces with equation
that are invariant under the action of the halfturn around Oz.
With the  sign, the halfturn swaps the two faces of the surface; taking the axis of the halfturn 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 halfturn around Oz that does not swap the faces, the two halfturns 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.
next surface  previous surface  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2017