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MÖBIUS SURFACE
See the "exam papers" X 1977 math 2 and agreg 1929. 
Cartesian parametrization: .
Toroidal equation: . Cylindrical equation: . Cartesian equation: , i.e. , simplifying by y, or also . Ruled cubic surface. Gaussian curvature: . Selfintersection line: x = a ; y = z ; axis of symmetry Ox. Directrix cone with directrix which is a clelia with parameter n = 1/2. 
The Möbius surface or halftwist surface is the nondevelopable ruled surface generated by the rotation of a line on a plane turning on itself around one of its lines with an angular speed equal to twice that of the line; it is therefore a special case of rotoid.
The Möbius surface is called this way because its portion obtained for with is a Möbius strip.
It can also be defined as the ruled surface the directrices
of which are a circle (here, ),
the axis of this circle (here, Oz) and a line forming an angle of
45° with respect to the plane of the circle, the projection of which
is a line tangent of the circle (here, x = a , y = z ). Since it
has two linear directrices, it is a conoidal
surface.

The respective intersection points between the generatrix and the circle, the red axis, and the green line, are . 
Besides, the Möbius surface is projectively equivalent
to Zindler's conoid; indeed
the change
transforms the homogeneous equation
of the Möbius surface into the equation
of this conoid.
The sections by the horizontal planes z = b are strophoids, with equation .  
The section by the sphere with center O and radius R is composed of a Viviani curve and of the equator of the sphere. 
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© Robert FERRÉOL 2018