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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: .
Self-intersection 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 half-twist surface is the non-developable 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.

Portion of the Möbius surface showing the evolution of a horizontal segment line to a vertical segment line.

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© Robert FERRÉOL  2018