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A second-order curve, i.e. the set of points in a (projective, affine or Euclidean) plane whose homogeneous coordinates (with respect to some projective, affine or Cartesian coordinate system) satisfy an equation of the second degree:

The symmetric bilinear form

is called the polar form of . Two points and for which are said to be polar conjugates with respect to the conic. If the line intersects the conic at points and if are polar conjugates with respect to the conic, then form a harmonic quadruple. The only self-conjugate points are the points of the conic itself. The pole of a given line with respect to a conic is the point that is polar conjugate with all the points of the line. The set of points in the plane that are polar conjugate with a given point with respect to a conic is called the polar of with respect to the conic. The polar of is defined by the linear equation in the coordinates . If , the polar of is a straight line; if , the polar of is the whole plane. In this case lies on the conic and is called a singular point of the conic. If , the conic has no singular points and is said to be non-degenerate or to be non-decomposing (non-splitting). In the projective plane this is a real or imaginary oval. A non-degenerate conic defines a correlation on the projective plane, i.e. a bijective mapping from the set of points onto the set of lines. A tangent to a non-degenerate conic is the polar of the point of tangency. If , the conic is a pair of real or imaginary straight lines intersecting at a singular point. If , every point of the conic is singular and the conic itself is a pair of coincident real straight lines (a double line). The affine properties of a conic are distinguished by the specific nature of its location and by the points and lines associated with it with respect to the distinguished line — the line at infinity. A conic is of hyperbolic, elliptic or parabolic type according to whether it intersects the line at infinity , does not intersect it or is tangent to it . Here

The centre of a conic is the pole of the line at infinity, a diameter is the polar of a point at infinity, an asymptote is a tangent to the conic at a point at infinity. Two diameters are conjugate with respect to the conic if their points at infinity are polar conjugates with respect to the conic.

The metric properties of a conic are determined from its affine properties by the invariance of the distance between two arbitrary points. The diameter of a conic that is orthogonal to the conjugate diameter is an axis of symmetry of the conic and is called an axis. A directrix of a conic is the polar of a focus.


[1] S.P. Finikov, "Analytic geometry" , Moscow (1952) (In Russian)
[2] N.V. Efimov, "A short course of analytical geometry" , Moscow (1967) (In Russian)



[a1] G. Salmon, "A treatise on conic sections" , Longman (1879)
[a2] O. Giering, "Vorlesungen über höhere Geometrie" , Vieweg (1982)
How to Cite This Entry:
Conic. V.S. Malakhovskii (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098