# Quadric

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A quadric is a surface of the second order. In a three-dimensional (projective, affine or Euclidean) space a quadric is a set of points whose homogeneous coordinates $x _ {0} , x _ {1} , x _ {2} , x _ {3}$( with respect to a projective, affine or Cartesian system of coordinates) satisfy a homogeneous equation of degree two:

$$F ( x) \equiv \ \sum _ {i , j = 0 } ^ { 3 } a _ {ij} x _ {i} x _ {j} = 0 ,\ \ a _ {ij} = a _ {ji} .$$

The bilinear symmetric form

$$\Phi ( x , \widetilde{x} ) = \ \sum _ {i , j = 0 } ^ { 3 } a _ {ij} x _ {i} \widetilde{x} _ {j}$$

is called the polar form relative to $F ( x)$. Two points $M ^ \prime ( x _ {0} ^ \prime , x _ {1} ^ \prime , x _ {2} ^ \prime , x _ {3} ^ \prime )$, $M ^ {\prime\prime} ( x _ {0} ^ {\prime\prime} , x _ {1} ^ {\prime\prime} , x _ {2} ^ {\prime\prime} , x _ {3} ^ {\prime\prime} )$ at which $\Phi ( x ^ \prime , x ^ {\prime\prime} ) = 0$ are called conjugate points with respect to the quadric. If the line $M ^ \prime M ^ {\prime\prime}$ intersects the quadric at the points $N _ {1} , N _ {2}$ and the points $M ^ \prime , M ^ {\prime\prime}$ are conjugate to each other with respect to the quadric, then $N _ {1} , N _ {2}$ and $M ^ \prime , M ^ {\prime\prime}$ form a harmonic quadruple. The points of a quadric and only these are self-conjugate. A line all points of which lie on a quadric is called a generator of the quadric. By the pole of a given plane with respect to a quadric is meant the point that is conjugate to every point of this plane. The set of points in the space that are conjugate to a given point $M ^ \prime$ with respect to a quadric is called the polar of $M ^ \prime$ with respect to the quadric. A tangent plane to a quadric is the polar of the point of contact. The polar of a point $M ^ \prime$ is defined by the linear equation $\Phi ( x , x ^ \prime ) \equiv 0$ with respect to the coordinates $x _ {0} , x _ {1} , x _ {2} , x _ {3}$. If $\Phi ( x , x ^ \prime ) \not\equiv 0$, then the polar of $M ^ \prime$ is a plane; if $\Phi ( x , x ^ \prime ) \equiv 0$, then the polar of $M ^ \prime$ is the whole space. In this case $M ^ \prime$ belongs to the quadric and is called a singular point of it. If $R = \mathop{\rm rank} ( a _ {ij} ) = 4$, then the quadric has no singular points and is called a non-degenerate quadric. In projective space this is an imaginary ovaloid, a real ovaloid or a ruled quadric. A non-degenerate quadric determines a correlation (or polarity), i.e. a bijective mapping from the set of points of projective space onto the set of planes. A ruled non-degenerate quadric has two distinct families of generators, distributed on the quadric so that any two lines of the same family are non-intersecting, while two lines of different families intersect at one point. If $R = 3$, then a quadric is a cone (real or imaginary) with vertex at the unique singular point. A real cone has a single family of generators, passing through its vertex. If $R = 2$, then the quadric splits into a pair of planes (real or imaginary), intersecting in a line consisting of its singular points. If $R = 1$, then a quadric is a double real plane consisting of singular points. The affine properties of a quadric are distinguished by its behaviour with respect to the plane at infinity, $x _ {0} = 0$. For example, an ellipsoid (hyperboloid, paraboloid) is a non-degenerate quadric that does not intersect (does intersect, is tangent to) the plane at infinity. The centre of a quadric is the pole of the plane at infinity, a diameter is a line through the centre.

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How to Cite This Entry:
Quadric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadric&oldid=48366
This article was adapted from an original article by V.S. Malakhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article