A general plane quartic curve is a curve of the form


Examples include the ampersand curve, bean curve, bicorn,bicuspid curve, bifoliate, bifolium, bow, bullet nose, capricornoid, Cartesian ovals, Cassini ovals, and cruciform.
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 seven-dimensional 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 six-dimensional space (Du Val 1931). A similar but less complete relation exists between the tritangent planes of the canonical curve of genus 4 and an eight-dimensional polytope (Du Val 1931).
The maximum number of double points for a nondegenerate quartic curve is three.

A quartic curve of the form

can be written

and so is cubic in the coordinates


This transformation is a birational transformation.

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


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 and


In Figure (c), the tangent at P intersects the curve at W. Then

Finally, in Figure (d), the intersections of the tangents at P and Q are W and X. Then

(Honsberger 1991).