Hippopede

In geometry, a hippopede (from ἱπποπέδη meaning "horse fetter" in ancient Greek) is a plane curve determined by an equation of the form

(x² + y²)² = cx² + dy²,

where it is assumed that c>0 and c>d since the remaining cases either reduce to a single point or can be put into the given form with a rotation. Hippopedes are bicircular rational algebraic curves of degree 4 and symmetric with respect to both the x and y axes. When d>0 the curve has an oval form and is often known as an oval of Booth, and when d<0 the curve resembles a sideways figure eight, or lemniscate, and is often known as a lemniscate of Booth, after James Booth (1810–1878) who studied them. Hippopedes were also investigated by Proclus (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus. For d = −c, the hippopede corresponds to the lemniscate of Bernoulli.

Definition as spiric sections

Hippopedes with a = 1, b = 0.1, 0.2, 0.5, 1.0, 1.5, and 2.0.

Hippopedes with b = 1, a = 0.1, 0.2, 0.5, 1.0, 1.5, and 2.0.

Hippopedes can be defined as the curve formed by the intersection of a torus and a plane, where the plane is parallel to the axis of the torus and tangent to it on the interior circle. Thus it is a spiric section which in turn is a type of toric section.

If a circle with radius a is rotated about an axis at distance b from its center, then the equation of the resulting hippopede in polar coordinates

$r^2 = 4 b (a- b \sin^{2} \theta)\,$

or in Cartesian coordinates

(x² + y²)² + 4b(b − a)(x² + y²) = 4b²x².

Note that when a>b the torus intersects itself, so it does not resemble the usual picture of a torus.

See also

• List of curves

References

• Lawrence JD. (1972) Catalog of Special Plane Curves, Dover. Pp. 145–146.
• Booth J. A Treatise on Some New Geometrical Methods, Longmans, Green, Reader, and Dyer, London, Vol. I (1873) and Vol. II (1877).
• Weisstein, Eric W., "Hippopede" from MathWorld.
• "Hippopede" at 2dcurves.com
• "Courbes de Booth" at Encyclopédie des Formes Mathématiques Remarquables

External links

• "The Hippopede of Proclus" at The National Curve Bank

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Algebraic curves Spiric sections Geometry stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем написать реферат Hessian curve Ferdinand Albin Pax Look at other dictionaries: Eudoxos von Knidos — (griechisch Εὔδοξος; * wohl zwischen 397 und 390 v. Chr. in Knidos, Kleinasien; † wohl zwischen 345 und 338 v. Chr. in Knidos) war ein berühmter griechischer Mathematiker, Astronom, Geograph, Arzt, Philosoph und Gesetzgeber der Antike. Seine …   Deutsch Wikipedia Eudoxus of Cnidus — (Greek Εὔδοξος ὁ Κνίδιος) (410 or 408 BC ndash; 355 or 347 BC) was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus s poem… …   Wikipedia List of curves — This is a list of curves, by Wikipedia page. See also list of curve topics, list of surfaces, Riemann surface. Algebraic curves*Cubic plane curve *Quartic plane curve *Quintic plane curve *Sextic plane curveRational curves*Ampersand curve… …   Wikipedia Otto E. Neugebauer — Born May 26, 1899(1899 05 26) Innsbruck, Austria Died February 19, 1990(1990 02 19) (aged 90) Citizenship United States Spouse Gre …   Wikipedia List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… …   Wikipedia Lemniscate of Booth — In algebraic geometry, the lemniscate of Booth, also called the Hippopede of Proclus , is a plane algebraic curve of degree four and genus zero, with equation:(x^2+y^2)^2 + 4y^2 = 4c(x^2+y^2),.It has an ordinary double point at the origin, and… …   Wikipedia Spiric section — A spiric section is a special case of a toric section, which is the intersection of a plane with a torus (σπειρα in ancient Greek). Spiric sections are toric sections in which the intersecting plane is parallel to the rotational symmetry axis of… …   Wikipedia Toric section — A toric section is an intersection of a plane with a torus, just as a conic section is the intersection of a plane with a cone. Mathematical formulaeIn general, toric sections are fourth order (quartic) plane curvesof the form:left( x^{2} + y^{2} …   Wikipedia Courbe quartique — En géométrie, une courbe quartique est une courbe algébrique de degré quatre. Elle peut être définie par une équation de degré quatre: Ax4 + By4 + Cx3y + Dx2y2 + Exy3 + Fx3 + Gy3 + Hx2y + Ixy2 + Jx2 + Ky2 + Lxy + Mx + Ny + P = 0. Cette équation a …   Wikipédia en Français Lemniscate De Booth — Lemniscates de Booth, pour différentes valeurs de c. c = 0,25 (noir) c = 0,5 (rouge) c = 0,75 (vert) c = 1 (bleu). En géométrie algébrique, la lemniscate de Booth, aussi appelée courbe de Booth, ovale de Booth ou encore hippopède de Proclus, est… …   Wikipédia en Français 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”