|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
|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
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.
© Robert FERRÉOL 2018