Conic

A second-order curve, i.e. the set of points in a (projective, affine or Euclidean) plane whose homogeneous coordinates $ x _ {0} , x _ {1} , x _ {2} $( with respect to some projective, affine or Cartesian coordinate system) satisfy an equation of the second degree:

$$ F ( x) \equiv \ \sum _ {i, j = 0 } ^ { 2 } a _ {ij} x _ {i} x _ {j} = 0,\ \ a _ {ij} = a _ {ji} . $$

The symmetric bilinear form

$$ \Phi ( x, \widetilde{x} ) = \ \sum _ {i, j = 0 } ^ { 2 } a _ {ij} x _ {i} \widetilde{x} _ {j} $$

is called the polar form of $ F ( x) $. Two points $ M ^ { \prime } = ( x _ {0} ^ \prime , x _ {1} ^ \prime , x _ {2} ^ \prime ) $ and $ M ^ { \prime\prime } = ( x _ {0} ^ {\prime\prime} , x _ {1} ^ {\prime\prime} , x _ {2} ^ {\prime\prime} ) $ for which $ \Phi ( x ^ \prime , x ^ {\prime\prime} ) = 0 $ are said to be polar conjugates with respect to the conic. If the line $ M ^ { \prime } M ^ { \prime\prime } $ intersects the conic at points $ N _ {1} , N _ {2} $ and if $ M ^ { \prime } , M ^ { \prime\prime } $ are polar conjugates with respect to the conic, then $ N _ {2} , N _ {2} , M ^ { \prime } , M ^ { \prime\prime } $ 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 $ M ^ { \prime } $ with respect to a conic is called the polar of $ M ^ { \prime } $ with respect to the conic. The polar of $ M ^ { \prime } $ is defined by the linear equation $ \Phi ( x, x ^ \prime ) = 0 $ in the coordinates $ x _ {0} , x _ {1} , x _ {2} $. If $ \Phi ( x, x ^ \prime ) \not\equiv 0 $, the polar of $ M ^ { \prime } $ is a straight line; if $ \Phi ( x, x ^ \prime ) \equiv 0 $, the polar of $ M ^ { \prime } $ is the whole plane. In this case $ M ^ { \prime } $ lies on the conic and is called a singular point of the conic. If $ R = \mathop{\rm rank} ( a _ {ij} ) = 3 $, 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 $ R = 2 $, the conic is a pair of real or imaginary straight lines intersecting at a singular point. If $ R = 1 $, 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 $ x _ {0} = 0 $— the line at infinity. A conic is of hyperbolic, elliptic or parabolic type according to whether it intersects the line at infinity $ ( \delta < 0) $, does not intersect it $ ( \delta > 0) $ or is tangent to it $ ( \delta = 0) $. Here

$$ \delta = \ \left | \begin{array}{ll} a _ {11} &a _ {12} \\ a _ {21} &a _ {22} \\ \end{array} \ \right | . $$

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.

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