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STRICTION LINE OF A NON DEVELOPABLE RULED SURFACE
Parametrization: ,
for the ruled surface
represented by ;
If we normalize (), the formula simplifies to and then M_{1} describes the striction line iff . The distribution parameter: can be geometrically interpreted by the formula , where is the angle between the tangent plane at C and the tangent plane at M (Chasles rule); in an equivalent fashion, the surface generated by the normals along the generatrix is a rectangular hyperbolic paraboloid with cylindrical equation in a frame Cxyz, Cz being the generatrix. As a consequence, when d >
0 (righthanded surface), the tangent plane turns counterclockwise as we
move along the generatrix, and clockwise in the lefthanded case.

The striction line of a non developable
ruled surface is the
locus of the central points (or striction points) of each
non parabolic generatrix of the surface; these points are
 the points where the tangent plane
is perpendicular to the asymptotic plane (i.e. the limit of the tangent
plane when it goes to infinity along the generatrix), or,
 the points where the absolute value
of the total curvature along a generatrix has a maximum (see the above
Lamarle formula), or even
 the limit points of the common perpendicular
to the generatrix and another generatrix that approaches it.
The latter definition also works in the case where the
surface is developable, and the resulting line is none other than the cuspidal
edge of this surface.
The striction line is not necessarily perpendicular to the generatrices (the condition is the orthogonality of M_{1}' and a', not that of M_{1}' and a). However, it contains (locally?) singular points of the surface.
When the surface is ruled and has a directrix plane, the projection on the striction line of the directrix plane is the envelope of the projections of the generatrices.
Examples:
 the striction line of a conoid
is its axis in the right case, but not necessarily in the oblique case,
 the striction line of a onesheeted
hyperboloid is its bore circle when it is of revolution, but not in
the general case,
 the striction line of a ruled
helicoid is its bore helix,
 the striction line of the hyperbolic
paraboloid,
 the striction line of the Möbius
surface,
 the striction line of the milk
carton.
Some illustrations of these notions, by Robert March.

References: (Mir
p.64, 67) (Deltheil p.251) (Bouasse p. 514)
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© Robert FERRÉOL , Robert MARCH 2018