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: . 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.