Quartic Curve

A general plane quartic curve is a curve of the form

(1)

The incidence relations of the 28 bitangents of the general quartic curve can be put into a One-to-One correspondence with the vertices of a particular Polytope in 7-D space (Coxeter 1928, Du Val 1931). This fact is essentially similar to the discovery by Schoutte (1910) that the 27 Solomon's Seal Lines on a Cubic Surface can be connected with a Polytope in 6-D space (Du Val 1931). A similar but less complete relation exists between the tritangent planes of the canonical curve of genus 4 and an 8-D Polytope (Du Val 1931).

The maximum number of Double Points for a nondegenerate quartic curve is three.

A quartic curve of the form

$\begin{displaymath} y^2=(x-\alpha)(x-\xi)(x-\gamma)(x-\delta) \end{displaymath}$

(2)

can be written

$\begin{displaymath} \left({y\over x-\alpha}\right)^2=\left({1-{\beta-\alpha\over... ...lpha}}\right) \left({1-{\delta-\alpha\over x-\alpha}}\right), \end{displaymath}$

(3)

and so is Cubic in the coordinates

$\displaystyle X$	$\textstyle =$	$\displaystyle {1\over x-\alpha}$	(4)
$\displaystyle Y$	$\textstyle =$	$\displaystyle {y\over x-\alpha^2}.$	(5)

This transformation is a Birational Transformation.

$\begin{figure}\begin{center}\BoxedEPSF{Quartic.epsf scaled 831}\end{center}\end{figure}$

Let and be the Inflection Points and and the intersections of the line with the curve in Figure (a) above. Then

$\displaystyle A$	$\textstyle =$	$\displaystyle C$	(6)
$\displaystyle B$	$\textstyle =$	$\displaystyle 2A.$	(7)

In Figure (b), let

be the double tangent, and

the point on the curve whose

coordinate is the average of the

coordinates of

and

. Then $UV\vert\vert PQ\vert\vert RS$ and

$\displaystyle D$	$\textstyle =$	$\displaystyle F$	(8)
$\displaystyle E$	$\textstyle =$	$\displaystyle \sqrt{2}\,D.$	(9)

In Figure (c), the tangent at

intersects the curve at

. Then

$\begin{displaymath} G=8B. \end{displaymath}$

(10)

Finally, in Figure (d), the intersections of the tangents at

and

are

and

. Then

$\begin{displaymath} H=27B \end{displaymath}$

(11)

(Honsberger 1991).

References

Coxeter, H. S. M. ``The Pure Archimedean Polytopes in Six and Seven Dimensions.'' Proc. Cambridge Phil. Soc. 24, 7-9, 1928.

Du Val, P. ``On the Directrices of a Set of Points in a Plane.'' Proc. London Math. Soc. Ser. 2 35, 23-74, 1933.

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 114-118, 1991.

Schoutte, P. H. ``On the Relation Between the Vertices of a Definite Sixdimensional Polytope and the Lines of a Cubic Surface.'' Proc. Roy. Akad. Acad. Amsterdam 13, 375-383, 1910.