In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a hippopede () is a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...
determined by an equation of the form
:
where it is assumed that and 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
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
,
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s of
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
4 and symmetric with respect to both the and axes.
Special cases
When ''d'' > 0 the curve has an oval form and is often known as an oval of Booth, and when the curve resembles a sideways figure eight, or
lemniscate
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternative ...
, and is often known as a lemniscate of Booth, after 19th-century mathematician
James Booth
James Booth (born David Noel Geeves; 19 December 1927 – 11 August 2005) was an English film, stage and television actor and screenwriter. Though considered handsome enough to play leading roles, and versatile enough to play a wide variety ...
who studied them. Hippopedes were also investigated by
Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
(for whom they are sometimes called Hippopedes of Proclus) and
Eudoxus. For , the hippopede corresponds to the
lemniscate of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. I ...
.
Definition as spiric sections
Hippopedes can be defined as the curve formed by the intersection of a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
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
In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form
:(x^2+y^2)^2=dx^2+ey^2+f. \,
Equivalently, spiric sections can be defined as bicircular quartic curves that are symme ...
which in turn is a type of
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. Special cases have been known since antiquity, and the general case was studied by Jean Gaston Darboux..
Mathematical ...
.
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 coordinate
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the ...
s
:
or in
Cartesian coordinate
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
s
:
.
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
This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), physics, engineering, economics, medicine, biology, psychology, ecology, etc.
Mathematics (Geometry) ...
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).
*{{MathWorld, title=Hippopede, urlname=Hippopede
"Hippopede" at 2dcurves.com"Courbes de Booth" at Encyclopédie des Formes Mathématiques Remarquables
External links
Algebraic curves
Spiric sections