Hippopede

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title: "Hippopede" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["quartic-curves", "spiric-sections"] description: "Class of quartic plane curves" topic_path: "general/quartic-curves" source: "https://en.wikipedia.org/wiki/Hippopede" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Class of quartic plane curves ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/8/89/PedalCurve1.gif" caption="Hippopede (red) given as the [[pedal curve]] of an [[ellipse]] (black). The equation of this hippopede is: 4x^2 + y^2 = (x^2 + y^2)^2"] ::

In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, 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.

Special cases

When d 0 the curve has an oval form and is often known as an oval of Booth, and when {{nowrap|d

Definition as spiric sections

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/02/Hippopede02.svg" caption="Hippopedes with ''a'' = 1, ''b'' = 0.1, 0.2, 0.5, 1.0, 1.5, and 2.0."] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/0d/Hippopede01.svg" caption="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

:(x^2+y^2)^2+4b(b-a)(x^2+y^2)=4b^2x^2.

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

References

• Lawrence JD. (1972) Catalog of Special Plane Curves, Dover Publications. 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).
• "Hippopede" at 2dcurves.com
• "Courbes de Booth" at Encyclopédie des Formes Mathématiques Remarquables

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::