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.

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.

References

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.

Return to Gallery

http://users.math.uoc.gr/~pamfilos/

Quartic

See Also

References