A bitangent is a line that is tangent to a curve at two distinct points.

Aa general plane quartic curve has 28 bitangents in the complex projective plane. However, as shown by Plücker (1839), the number of real bitangents of a quartic must be 28, 16, or a number less than 9. Plücker (Plücker 1839, Gray 1982) constructed the first as


(correcting the typo of (y+xy) for (x+y)) for k small and positive. Without mentioning its origin or significance, this curve with k=0 is termed the ampersand curve by Cundy and Rowlett (1989, p. 72).

As noted by Gray (1982), "the 28 bitangents became, and remain, a topic of delight."


Trott (1997) subsequently gave the beautiful symmetric quartic curve with 28 real bitangents


which is illustrated above.

See also

Ampersand Curve, Bitangent Vector, Klein's Equation, Plücker Characteristics, Secant Line, Solomon's Seal Lines, Tangent Line

Explore with Wolfram|Alpha


Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989.Frame, J. S. "The Classes and Representations of the Groups of 27 Lines and 28 Bitangents." Ann. Mat. Pura Appl. 32, 83-119, 1951.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: Adolph Marcus, 1839.Shioda, T. "Weierstrass Transformations and Cubic Surfaces." Comm. Math. Univ. Sancti Pauli 44, 109-128, 1995.Trott, M. "Applying GroebnerBasis to Three Problems in Geometry." Mathematica Educ. Res. 6, 15-28, 1997.

Cite this as:

Weisstein, Eric W. "Bitangent." From MathWorld--A Wolfram Web Resource.

Subject classifications