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SALMON QUARTIC
Curve studied by Salmon in 1852 (see higher
plane curves p. 45)
George Salmon (18191904): 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 "ringshaped". 
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. 

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