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 circular algebraic curve is a type of
plane 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 ...
determined by an equation ''F''(''x'', ''y'') = 0, where ''F'' is a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
with real coefficients and the highest-order terms of ''F'' form a polynomial divisible by ''x''
2 + ''y''
2. More precisely, if
''F'' = ''F''
''n'' + ''F''
''n''−1 + ... + ''F''
1 + ''F''
0, where each ''F''
''i'' is
homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
of degree ''i'', then the curve ''F''(''x'', ''y'') = 0 is circular if and only if ''F''
''n'' is divisible by ''x''
2 + ''y''
2.
Equivalently, if the curve is determined in
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
by ''G''(''x'', ''y'', ''z'') = 0, where ''G'' is a homogeneous polynomial, then the curve is circular if and only if ''G''(1, ''i'', 0) = ''G''(1, −''i'', 0) = 0. In other words, the curve is circular if it contains the
circular points at infinity, (1, ''i'', 0) and (1, −''i'', 0), when considered as a curve in the
complex projective plane
In mathematics, the complex projective plane, usually denoted or is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2, ...
.
Multicircular algebraic curves
An algebraic curve is called ''p''-circular if it contains the points (1, ''i'', 0) and (1, −''i'', 0) when considered as a curve in the complex projective plane, and these points are singularities of order at least ''p''. The terms ''bicircular'', ''tricircular'', etc. apply when ''p'' = 2, 3, etc. In terms of the polynomial ''F'' given above, the curve ''F''(''x'', ''y'') = 0 is ''p''-circular if ''F''
''n''−''i'' is divisible by (''x''
2 + ''y''
2)
''p''−''i'' when ''i'' < ''p''. When ''p'' = 1 this reduces to the definition of a circular curve. The set of ''p''-circular curves is invariant under
Euclidean transformations. Note that a ''p''-circular curve must have degree at least 2''p''.
When ''k'' is 1 this says that the set of lines (0-circular curves of degree 1) together with the set of circles (1-circular curves of degree 2) form a set which is invariant under inversion.
Examples
* The
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
is the only circular conic.
*
Conchoids of de Sluze (which include several well-known cubic curves) are circular cubics.
*
Cassini ovals (including 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 ...
),
toric sections and
limaçon
In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. I ...
s (including the
cardioid) are bicircular quartics.
*
Watt's curve
In mathematics, Watt's curve is a circular algebraic curve, tricircular algebraic curve, plane algebraic curve of sextic, degree six. It is generated by two circles of radius ''b'' with centers distance 2''a'' apart (taken to be at (±''a'', 0)) ...
is a tricircular sextic.
Footnotes
References
*
"Courbe Algébrique Circulaire" at Encyclopédie des Formes Mathématiques Remarquables* {{in lang, fr}
"Courbe Algébrique Multicirculaire" at Encyclopédie des Formes Mathématiques Remarquables
Curves
Analytic geometry