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KLEIN QUARTIC
Curve studied by Klein in 1879 (Uber
die Transformationen siebenter Ordnung der elliptischen Funktionen, Math.
Ann. 14 (1879), 428-471. Œuvres, Tome III, p. 90-136).
Félix Klein (1849-1925): German mathematician. Webography: en.wikipedia.org/wiki/Klein_quartic mathworld.wolfram.com/KleinQuartic.html math.univ-lyon1.fr/~germoni/memoires/quartique.pdf |
1) The affine Klein quartic:
Cartesian equation:
where
so that pqr = 0 is the reunion of 3 lines forming an equilateral triangle centred on O. Quartic of genus 3. Polar equation: . REMARK 1: the quartic passes by the three intersection points between
the circle
and the previous 3 lines and is tangent (doubly) to these 3 lines at these
3 points.
Compare with the definition of the Kummer surface. |
Case , with an illustration of the remark 1. |
The same curve, represented along with its Hessian, intersecting the curve at its inflexion points. When , which is the case here, the tangents at the inflexion points are parallel to a symmetry axis. |
The (affine) Klein quartic is the curve above in the
special case where .
Its particularity is that its tangents at the 6 inflexion points (in green opposite) pass through another inflexion point and form two equilateral triangles. Compare to the Loriga quartic (which has the same property, but is different). |
|
2) The projective Klein quartic (projectively equivalent
to the previous one):
Homogeneous equation: |
View of the cone
(that can be referred to as the "Klein cone") cut by the plane ,
providing an affine realisation of the projective Klein quartic (different
from that of the first paragraph).
The cylindrical equation of this cone in a frame where Oz is the axis of rotation is: . |
Klein cone, by Alain Esculier
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© Robert FERRÉOL
2017