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Ampersand Curve

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AmpersandCurve

The ampersand curve is the name given by Cundy and Rollett (1989, p. 72) to the quartic curve with implicit equation

(y^2-x^2)(x-1)(2x-3)=4(x^2+y^2-2x)^2.
(1)

Although it is not mentioned by Cundy and Rollett, this curve is significant because it is the original example (after subtracting a small positive constant k) of a quartic curve having 28 real bitangents constructed by Plücker (Plücker 1839, Gray 1982), namely Plücker's quartic.

The ampersand curve has crunodes at (1,-1), (0,0), and (1,1).

The points with horizontal tangent lines are at (1/2,+/-1/2sqrt(5)), (1/(120)(159-sqrt(201)),+/-1/(40)sqrt(1389+67sqrt((67)/3))), and (1/(120)(159+sqrt(201)),+/-1/(40)sqrt(1389-67sqrt((67)/3))). The points with vertical tangent lines are at (-1/(10),+/-1/(10)sqrt(23)) and (3/2,+/-1/2sqrt(3)).

The polar equation is given by solving the quadratic equation

r^2[2cos(2theta)+cos(4theta)+9]-r[37costheta+5cos(3theta)] +[22cos(2theta)+16]=0.
(2)

The area enclosed by the ampersand is given approximately by

A approx 1.06656
(3)

(OEIS A101801) and the perimeter approximately by

s approx 9.19756
(4)

(OEIS A193670).

See also

Plücker's Quartic

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References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989.Gray, J. "From the History of a Simple Group." Math. Intell. 4, 59-67, 1982. Reprinted in The Eightfold Way: The Beauty of the Klein Quartic (Ed. S. Levy). New York: Cambridge University Press, pp. 115-131, 1999.Plücker, J. Theorie der algebraischen Curven: Gegründet auf eine neue Behandlungsweise der analytischen Geometrie. Berlin, Germany: Adolph Marcus, 1839.Sloane, N. J. A. Sequences A101801 and A193670 in "The On-Line Encyclopedia of Integer Sequences."

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Ampersand Curve

Cite this as:

Weisstein, Eric W. "Ampersand Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AmpersandCurve.html

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