Circular algebraic curve

Circular algebraic curve

In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation F(xy) = 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 FFn + Fn−1 + ... + F1 + F0, where each Fi is homogeneous of degree i, then the curve F(xy) = 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.

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(xy) = 0 is p-circular if Fni is divisible by (x2 + y2)pi 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 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

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Circular — is a basic geometric shape such as a Circle. Contents 1 Documents 2 Travel and transportation 3 Places …   Wikipedia

  • Circular points at infinity — In projective geometry, the circular points at infinity in the complex projective plane (also called cyclic points or isotropic points) are (1: i: 0) and (1: −i: 0). Here the coordinates are homogeneous coordinates (x: y: z); so that the line at… …   Wikipedia

  • Curve fitting — best fit redirects here. For placing ( fitting ) variable sized objects in storage, see fragmentation (computer). Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,… …   Wikipedia

  • Cubic plane curve — A selection of cubic curves. See information page for details. Cubic curve redirects here. For information on polynomial functions of degree 3, see Cubic function. In mathematics, a cubic plane curve is a plane algebraic curve C defined by a… …   Wikipedia

  • List of circle topics — This list of circle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or concretely in physical space. It does not include metaphors like inner circle or circular reasoning in… …   Wikipedia

  • Polar coordinate system — Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 …   Wikipedia

  • Conic section — Types of conic sections: 1. Parabola 2. Circle and ellipse 3. Hyperbola …   Wikipedia

  • Superellipse — The superellipse (or Lamé curve) is the geometric figure defined in the Cartesian coordinate system as the set of all points ( x , y ) with:left|frac{x}{a} ight|^n! + left|frac{y}{b} ight|^n! = 1 where n gt; 0 and a and b are the semi major and… …   Wikipedia

  • Tangent half-angle formula — In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable t . These identities are known collectively as the tangent half angle formulae …   Wikipedia

  • Eudoxus of Cnidus — (Greek Εὔδοξος ὁ Κνίδιος) (410 or 408 BC ndash; 355 or 347 BC) was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus s poem… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”