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 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 c ...
determined by an equation ''F''(''x'', ''y'') = 0, where ''F'' is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
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 often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
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. T ...
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 In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.
Coordinates ...
, (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 P2(C), 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 \mathbf^3,\qquad (Z_1, ...
.
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 points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
is the only circular conic.
*
Conchoids of de Sluze (which include several well-known cubic curves) are circular cubics.
*
Cassini oval
In geometry, a Cassini oval is a quartic plane curve defined as the locus (mathematics), locus of points in the plane (geometry), plane such that the Product_(mathematics), product of the distances to two fixed points (Focus (geometry), foci) is ...
s (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 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 ...
s 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. It c ...
s (including the
cardioid
In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spi ...
) are bicircular quartics.
*
Watt's curve
In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius ''b'' with centers distance 2''a'' apart (taken to be at (±''a'', 0)). A line segment of length 2''c'' attaches to a p ...
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