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SALMON QUARTIC

Curve studied by Salmon in 1852 (see higher plane curves p. 45)
George Salmon (1819-1904): Irish mathematician.

 
Cartesian equation: , a, b > 0 , or 
Cartesian parametrization: 
Quartic of genus 3 when b is different from a and from , decomposed quartic when b = a, and of genus 2 for b = .

 
For b = a , the quartic can be decomposed in two ellipses (cf the second equation)
For b < a, it is composed of 4 connected components surrounding the vertices of a square (with coordinates ), which is the maximum possible for a quartic.

For a < b, it is composed of 2 connected components, with the central component degenerating into an isolated point for b = .
The quartic is said to be "ring-shaped".
For b, there is only one component left.
All this can be easily seen on the surface  the contour lines of which are the Salmon quartic.
For 0 < b < a, the Salmon quartic has, visually, 4 times 6 = 24 bitangents, like 4 circles would.
Salmon quartic and 6 of its 24 bitangents
For , the 4 connected components have a concave part and therefore have one bitangent each.
The Salmon quartic has, in this case, 24+4=28 real bitangents, the maximum possible for a quartic.

The 4 additional bitangents in that case.

Yet, a quartic can only have 28, 16, or at most 8 real bitangents.
Where are the four missing bitangents in the case where the 4 components are convex?
They indeed are real (in blue opposite), but their tangency points with the quartic have complex coordinates!

 
 
To obtain the same phenomenon of the 4 components and the 28 bitangents, we can cross ellipses in order to emphasize the concavity of the "meniscus".
Opposite, the curve 
for b=0.3a and k=0.01, with its two directrix ellipses, and its 28 bitangents.
(Cf. the Trott curve).

Cf. also the Plücker quartic, which is the historical example of a quartic with 28 bitangents.
 
Salmon studied, more generally, the quartic 
the different shapes of which indicated below are visible on the plot of the surface  opposite.
b < a1 < a2 b=a1<a2 a1 < b < a2 a1 < a2 = b a1<a2<b<(a1^4+a2^4)^(1/4) a1<a2<(a1^4+a2^4)^(1/4)<=b

 
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© Robert FERRÉOL 2017