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Quartic Curve

A general plane quartic curve is a curve of the form
$ +Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+O=0.$

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

\end{displaymath} (2)

can be written
\left({y\over x-\alpha}\right)^2=\left({1-{\beta-\alpha\over...
\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 $P$ and $Q$ be the Inflection Points and $R$ and $S$ the intersections of the line $PQ$ with the curve in Figure (a) above. Then

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

In Figure (b), let $UV$ be the double tangent, and $T$ the point on the curve whose $x$ coordinate is the average of the $x$ coordinates of $U$ and $V$. 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 $P$ intersects the curve at $W$. Then
\end{displaymath} (10)

Finally, in Figure (d), the intersections of the tangents at $P$ and $Q$ are $W$ and $X$. Then
\end{displaymath} (11)

(Honsberger 1991).

See also Cubic Surface, Pear-Shaped Curve, Solomon's Seal Lines


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.

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© 1996-9 Eric W. Weisstein