In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a hippopede () is a
plane curve
In mathematics, a plane curve is a curve in a plane that may be 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 plane c ...
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 reason. 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 ...
,
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 cu ...
s of
degree 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 ( or ) is any of several figure-eight or -shaped curves. The word comes from the Latin , meaning "decorated with ribbons", from the Greek (), meaning "ribbon",. or which alternatively may refer to the wool fr ...
, 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. He is best known for his role as Private Henry Hook in '' Zulu.''
''Variety'' called him "a punchy b ...
who studied them. Hippopedes were also investigated by
Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...
(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 (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
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 symm ...
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 specifies a given point in a plane by using a distance and an angle as its two coordinates. These are
*the point's distance from a reference point called the ''pole'', and
*the point's direction from ...
s
:
or in
Cartesian coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
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
Quartic curves
Spiric sections