Circular algebraic curve
Encyclopedia
In geometry
, a circular algebraic curve is a type of plane algebraic curve determined by an equation F(x, y) = 0, where F is a polynomial
with real coefficients and the highest-order terms of F form a polynomial divisible by x2 + y2. More precisely, if
F = Fn + Fn−1 + ... + F1 + F0, where each Fi is homogeneous
of degree i, then the curve F(x, y) = 0 is circular if and only if Fn is divisible by x2 + y2.
Equivalently, if the curve is determined in homogeneous coordinates
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.
. Note that a p-circular curve must have degree at least 2p.
The set of p-circular curves of degree p + k, where p may vary but k is a fixed positive integer, is invariant under inversion. 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.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a circular algebraic curve is a type of plane algebraic curve determined by an equation F(x, y) = 0, where F is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
with real coefficients and the highest-order terms of F form a polynomial divisible by x2 + y2. More precisely, if
F = Fn + Fn−1 + ... + F1 + F0, where each Fi is homogeneous
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
of degree i, then the curve F(x, y) = 0 is circular if and only if Fn is divisible by x2 + y2.
Equivalently, if the curve is determined in homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
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
Circular points at infinity
In projective geometry, the circular points at infinity in the complex projective plane are and ....
, (1, i ,0) and (1, −i, 0), when considered as a curve in the complex projective plane.
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 Fn−i is divisible by (x2 + y2)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 transformationsEuclidean group
In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
. Note that a p-circular curve must have degree at least 2p.
The set of p-circular curves of degree p + k, where p may vary but k is a fixed positive integer, is invariant under inversion. 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 circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
is the only circular conic. - Conchoids of de SluzeConchoid of de SluzeThe conchoid of de Sluze is a family of plane curves studied in 1662 by René François Walter, baron de Sluze.The curves are defined by the polar equationr=\sec\theta+a\cos\theta \,....
(which includes several well-known cubic curves) are circular cubics. - Cassini ovalCassini ovalA Cassini oval is a plane curve defined as the set of points in the plane such that the product of the distances to two fixed points is constant. This is related to an ellipse, for which the...
s (including the lemniscate of BernoulliLemniscate of BernoulliIn geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus of points P so that PF1·PF2 = a2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscus, which is...
), toric sectionToric sectionA 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.-Mathematical formulae:In general, toric sections are fourth-order plane curves of the form...
s and limaçonLimaçonIn geometry, a limaçon or limacon , also known as a limaçon of Pascal, is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller...
s (including the cardioidCardioidA 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 is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp...
) are bicircular quartics. - Watt's curveWatt's curveIn 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 2a apart 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...
is a tricircular sextic.