A general plane quartic curve is a curve of the form

(1) 
The maximum number of Double Points for a nondegenerate quartic curve is three.
A quartic curve of the form
(2) 
(3) 
(4)  
(5) 
Let and be the Inflection Points and and the intersections of the line
with the curve in Figure (a) above. Then
(6)  
(7) 
(8)  
(9) 
(10) 
(11) 
See also Cubic Surface, PearShaped Curve, Solomon's Seal Lines
References
Coxeter, H. S. M. ``The Pure Archimedean Polytopes in Six and Seven Dimensions.'' Proc. Cambridge Phil. Soc. 24, 79, 1928.
Du Val, P. ``On the Directrices of a Set of Points in a Plane.'' Proc. London Math. Soc. Ser. 2 35, 2374, 1933.
Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 114118, 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, 375383, 1910.
© 19969 Eric W. Weisstein