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Quartic Curve

A general plane quartic curve is a curve of the form

 Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+O=0.
(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, folium, hippopede, Kampyle of Eudoxus, Klein quartic, knot curve, lemniscate, limaçon, links curve, pear-shaped curve, piriform, 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

 y^2=(x-alpha)(x-beta)(x-gamma)(x-delta)
(2)

can be written

 [y/((x-alpha)^2)]^2 =(1-(beta-alpha)/(x-alpha))(1-(gamma-alpha)/(x-alpha))(1-(delta-alpha)/(x-alpha)),
(3)

and so is cubic in the coordinates

X = 1/(x-alpha)
(4)
Y = y/((x-alpha)^2)
(5)

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


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

Quartic

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

A = C
(6)
B = 2A.
(7)

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 UV∥PQ∥RS and

D = F
(8)
E = sqrt(2)D.
(9)

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

 G=8B.
(10)

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

 H=27B
(11)

(Honsberger 1991).

SEE ALSO:Algebraic Curve, Cubic Curve, Cubic Surface, Pear-Shaped Curve, Quadratic Curve, Solomon's Seal Lines

REFERENCES:

Cassels, J. W. S. Ch. 8 in Lectures on Elliptic Curves. New York: Cambridge University Press, 1991.

Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil. Soc. 24, 7-9, 1928.

Du Val, P. "On the Directrices of a Set of points in a Plane." Proc. London Math. Soc. Ser. 2 35, 23-74, 1933.

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 114-118, 1991.

Salmon, G. A Treatise on Higher Plane Curves, Intended As a Sequel to a Treatise on Conic Sections, 3rd ed. Dublin: Hodges, 1879.

Schoute, 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, 375-383, 1910.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 49, 1991.


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