Bicorn

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

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.

Contents

History

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.

Properties

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

References


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Synonyms:

Look at other dictionaries:

  • 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 (monster) — Bicorn (Also known as Bicrone) is a mythological creature related to the Unicorn with two horns that has the reputation of devouring kind hearted and devoted husbands, and is thus plump and well fed. His counterpart is the Chichevache, which… …   Wikipedia

  • 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

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