View source for Conic

A second-order curve, i.e. the set of points in a (projective, affine or Euclidean) plane whose homogeneous coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249501.png" /> (with respect to some projective, affine or Cartesian coordinate system) satisfy an equation of the second degree:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249502.png" /></td> </tr></table>

The symmetric bilinear form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249503.png" /></td> </tr></table>

is called the polar form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249504.png" />. Two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249506.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249507.png" /> are said to be polar conjugates with respect to the conic. If the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249508.png" /> intersects the conic at points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c0249509.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495010.png" /> are polar conjugates with respect to the conic, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495012.png" /> with respect to a conic is called the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495013.png" /> with respect to the conic. The polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495014.png" /> is defined by the linear equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495015.png" /> in the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495017.png" />, the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495018.png" /> is a straight line; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495019.png" />, the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495020.png" /> is the whole plane. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495021.png" /> lies on the conic and is called a singular point of the conic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495022.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495023.png" />, the conic is a pair of real or imaginary straight lines intersecting at a singular point. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495024.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495025.png" /> — the line at infinity. A conic is of hyperbolic, elliptic or parabolic type according to whether it intersects the line at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495026.png" />, does not intersect it <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495027.png" /> or is tangent to it <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495028.png" />. Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024950/c02495029.png" /></td> </tr></table>

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Finikov,   "Analytic geometry" , Moscow  (1952)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.V. Efimov,   "A short course of analytical geometry" , Moscow  (1967)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Salmon,   "A treatise on conic sections" , Longman  (1879)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Giering,   "Vorlesungen über höhere Geometrie" , Vieweg  (1982)</TD></TR></table>