For the hat, see Bicorne.
For the mythical beast, see Bicorn (monster).

In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation

y2(a2x2) = (x2 + 2aya2)2.

It has two cusps and is symmetric about the y-axis.



In 1864, James Joseph Sylvester studied the curve

y4xy3 − 8xy2 + 36x2y + 16x2 − 27x3 = 0

in connection with the classification of quintic equations; he named the curve a bicorn because it has two cusps. This curve was further studied by Arthur Cayley in 1867.


The bicorn is a plane algebraic curve of degree four and genus zero. It has two cusp singularities in the real plane, and a double point in the complex projective plane at x=0, z=0 . If we move x=0 and z=0 to the origin substituting and perform an imaginary rotation on x bu substituting ix/z for x and 1/z for y in the bicorn curve, we obtain

(x^2-2az+a^2z^2)^2  = x^2+a^2z^2.\,

This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at x = ± i and z=1.

A transformed bicorn with a = 1


The parametric equations of a bicorn curve are:

x = asin(θ) and y = \frac{\cos^2(\theta) \left(2+\cos(\theta)\right)}{3+\sin^2(\theta)} with -\pi\le\theta\le\pi

See also


Wikimedia Foundation. 2010.


См. также в других словарях:

  • bicorn — BICÓRN, bicornuri, s.n. Pălărie bărbătească (de uniformă) cu două colţuri. – Din fr. bicorne. Trimis de paula, 21.06.2002. Sursa: DEX 98  bicórn s. n., pl. bicórnuri Trimis de siveco, 10.08.2004. Sursa: Dicţionar ortografic  BICÓRN bicornuri n …   Dicționar Român

  • Bicorn — Bi corn, Bicorned Bi corned, Bicornous Bi*cor nous, a. [L. bicornis; bis twice + cornu horn: cf. F. bicorne. Cf. {Bickern}.] Having two horns; two horned; crescentlike. [1913 Webster] …   The Collaborative International Dictionary of English

  • Bicorn — Bicorn, zweihörnig; bicorniger, Zweihörniger. Beiname des Bacchus …   Herders Conversations-Lexikon

  • bicorn — [bī kôr′nyo͞o itbī′kôrn΄] adj. [L bicornis < bi , BI 1 + cornu,HORN] 1. having two horns or hornlike parts 2. crescent shaped: Also bicornuate [bī kôr′nyo͞o it] …   English World dictionary

  • Bicorn — /buy kawrn/, n. (in early French and English literature) a mythical animal, usually depicted as a grotesquely fat beast, that existed solely by devouring virtuous husbands. Cf. Chichivache. [1375 1425; late ME Bycorne < MF < L bicornis BICORN] *… …   Universalium

  • bicorn — adj. [L. bis, two; cornu, horn] Bearing two horns; crescentlike …   Dictionary of invertebrate zoology

  • bicorn — /buy kawrn/, adj. Also, bicornate /buy kawr nit, nayt/, bicornuate /buy kawr nyooh it, ayt /, bicornuous. 1. Bot., Zool. having two horns or hornlike parts. 2. shaped like a crescent. n. 3. bicorne. [1745 55; < L bicornis, equiv. to bi BI 1 +… …   Universalium

  • bicorn — noun A plane curve having two cusps …   Wiktionary

  • bicorn — Synonyms and related words: Cynthian, S shaped, corniform, crescent, crescent shaped, crescentic, crescentiform, crescentlike, falcate, falciform, horn shaped, horned, lunar, lunate, luniform, lunular, menisciform, meniscoid, moon shaped,… …   Moby Thesaurus

  • bicorn — adj. having two horns, having two horn like parts …   English contemporary dictionary

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