Bitangents of a quartic

In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for which all 28 of these lines have real numbers as their coordinates and therefore belong to the Euclidean plane.

An explicit quartic with twenty-eight real bitangents was first given by Plücker (1839). As Plücker showed, the number of real bitangents of any quartic must be 28, 16, or a number less than 9. Another quartic with 28 real bitangents can be formed by the locus of centers of ellipses with fixed axis lengths, tangent to two non-parallel lines.

Shioda (1995) gave a different construction of a quartic with twenty-eight bitangents, formed by projecting a cubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the line at infinity in the projective plane.

Example

The Trott curve, another curve with 28 real bitangents, is the set of points (x,y) satisfying the degree four polynomial equation

144

(

x

4

+

y

4

)

−

225

(

x

2

+

y

2

)

+

350

x

2

y

2

+

81

=

0.

{\displaystyle \displaystyle 144(x^{4}+y^{4})-225(x^{2}+y^{2})+350x^{2}y^{2}+81=0.}

These points form a nonsingular quartic curve that has genus three and that has twenty-eight real bitangents.

Like the examples of Plücker and of Blum and Guinand, the Trott curve has four separated ovals, the maximum number for a curve of degree four, and hence is an M-curve. The four ovals can be grouped into six different pairs of ovals; for each pair of ovals there are four bitangents touching both ovals in the pair, two that separate the two ovals, and two that do not.