Bitangents of a quartic

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.

[2] 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.

The points and lines of the Fano plane that are disjoint from a non-incident point-line pair form a triangle, and the bitangents of a quartic have been considered as being in correspondence with the 28 triangles of the Fano plane.

[6] The 28 bitangents of a quartic also correspond to pairs of the 56 lines on a degree-2 del Pezzo surface,[5] and to the 28 odd theta characteristics.

The 27 lines on the cubic and the 28 bitangents on a quartic, together with the 120 tritangent planes of a canonic sextic curve of genus 4, form a "trinity" in the sense of Vladimir Arnold, specifically a form of McKay correspondence,[7][8][9] and can be related to many further objects, including E7 and E8, as discussed at trinities.

The Trott curve and seven of its bitangents. The others are symmetric with respect to 90° rotations through the origin.
The Trott curve with all 28 bitangents.