Quartic Curve

A general plane quartic curve is a curve of the form

(1)

Examples include the ampersand curve, bean curve, bicorn, bicuspid curve, bifoliate, bifolium, bitangent-rich curve, bow, bullet nose, butterfly curve, capricornoid, cardioid, Cartesian ovals, Cassini ovals, conchoid of Nicomedes, cruciform, deltoid, devil's curve, Dürer's conchoid, eight curve, fish curve, hippopede, Kampyle of Eudoxus, Kepler's Folium, Klein quartic, knot curve, lemniscate, limaçon, links curve, pear-shaped curve, piriform curve, swastika curve, trefoil curve, and trifolium.

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 1933). This fact is essentially similar to the discovery by Schoute (1910) that the 27 Solomon's seal lines on a cubic surface can be connected with a polytope in six-dimensional space (Du Val 1933). 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 1933).

The maximum number of double points for a nondegenerate quartic curve is three.

A quartic curve of the form

(2)

can be written

(3)

and so is cubic in the coordinates

			(4)
			(5)

(Cassels 1991). This transformation is a birational transformation.

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

			(6)
			(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 and

			(8)
			(9)

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

(10)

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

(11)

(Honsberger 1991).