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