[alogo] Quartic

This is the curve represented by the graph of a polynomial of degree four: f(x) = a*x4+b*x3+c*x2+d*x+e. The curve has the following properties:
[1] It is affine-symmetric with respect to the axis passing through the point B(x0,0), x0 = -b/(4*a).
[2] The symmetry occurs along the direction of the tangent [B1B2] at B'(x0,f(x0)).
[3] Tangents at points D', D'' symmetric along [B1B2] intersect on the BB' axis.
In the case the curve has two flexes (I, J in the picture below), where the second derivative vanishes, then:
[4] The line of flexes [IJ] is parallel to [B1B2].
[5] There is a bitangent [C1C2] also parallel to [B1B2].
[6] The line of flexes [IJ] divides the distance of [C1C2] to [B1B2] in ratio 4/5.
[7] The segments JK, IJ on the line of flexes are in golden-ratio proportion: JK/IJ = (sqrt(5)-1)/2.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]

First the affine symmetry here is an affine transformation leaving the axis BB' fixed and mapping every point K to L, such that KL is parallel to [B1B2] and the middle A of KL is on BB'.
The conclusions are easily proved by translating the origin at B' and dividing with the highest power coefficient. Then the curve admits the representation of the form g(x) = x4 + u*x2 + v*x.
In this representation line y = v*x coincides with [B1B2]. Everything follows from simple calculations and the fact that the relations do not change by applying to the function affine transformations of the form (x'=r*x, y'=s*y).
Note that condition u < 0, characterizes the existence of two flexes.
In addition, all quartics are affinely equivalent to one of the three normal forms of quartics: x4, x4+x2, x4-x2.

See Also



Aude, H. T. R. Notes on Quartic Curves The American Mathematical Monthly, Vol. 56, No. 3. (Mar., 1949), pp. 165-170.
Feeman, T. G. & Marrero O. Affine Transformations, Polynomials, and Proportionality The American Mathematical Monthly, vol. 108, No. 10(Dec., 2001), 972-975.

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