Quadric

For the computing company, see Quadrics.
Not to be confused with Quadratic or Quartic.

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in (D + 1)-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates x1, x2, ..., xD+1, the general quadric is defined by the algebraic equation[1]

\sum_{i,j=1}^{D+1} x_i Q_{ij} x_j + \sum_{i=1}^{D+1} P_i x_i + R = 0

which may be compactly written in vector and matrix notation as:

x Q x^\mathrm{T} + P x^\mathrm{T} + R = 0\,

where x = (x1, x2, ..., xD+1) is a row vector, xT is the transpose of x (a column vector), Q is a (D + 1) × (D + 1) matrix and P is a (D + 1)-dimensional row vector and R a scalar constant. The values Q, P and R are often taken to be over real numbers or complex numbers, but a quadric may be defined over any ring. In general, the locus of zeros of a set of polynomials is known as an algebraic set, and is studied in the branch of algebraic geometry.

A quadric is thus an example of an algebraic set. For the projective theory see Quadric (projective geometry).

Euclidean plane and space

Quadrics in the Euclidean plane are those of dimension D = 1, which is to say that they are curves. Such quadrics are the same as conic sections, and are typically known as conics rather than quadrics.

Ellipse (e = 1/2), parabola (e=1) and hyperbola (e = 2) with fixed focus F and directrix.

In Euclidean space, quadrics have dimension D = 2, and are known as quadric surfaces. By making a suitable Euclidean change of variables, any quadric in Euclidean space can be put into a certain normal form by choosing as the coordinate directions the principal axes of the quadric. In three-dimensional Euclidean space there are 16 such normal forms.[2] Of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include planes, lines, points or even no points at all.[3] Quadric with nonzero Gaussian curvature is Darboux surface in three-dimensional Euclidean space.[4]

Non-degenerate real quadric surfaces
    Ellipsoid {x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,
    Spheroid (special case of ellipsoid) {x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,
    Sphere (special case of spheroid) {x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1 \,
    Elliptic paraboloid {x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,
    Circular paraboloid (special case of elliptic paraboloid) {x^2 \over a^2} + {y^2 \over a^2} - z = 0 \,
    Hyperbolic paraboloid {x^2 \over a^2} - {y^2 \over b^2} - z = 0 \,
   Elliptic hyperboloid of one sheet {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,
   Elliptic hyperboloid of two sheets {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = - 1 \,
Degenerate quadric surfaces
    Elliptic cone {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,
    Circular cone (special case of cone) {x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = 0 \,
    Elliptic cylinder {x^2 \over a^2} + {y^2 \over b^2} = 1 \,
    Circular cylinder (special case of elliptic cylinder) {x^2 \over a^2} + {y^2 \over a^2} = 1 \,
    Hyperbolic cylinder {x^2 \over a^2} - {y^2 \over b^2} = 1 \,
    Parabolic cylinder x^2 + 2ay = 0 \,

Projective geometry

The quadrics can be treated in a uniform manner by introducing homogeneous coordinates on a Euclidean space, thus effectively regarding it as a projective space. Thus if the original (affine) coordinates on RD+1 are

(x_1,\dots,x_{D+1})

one introduces new coordinates on RD+2

[X_0,\dots,X_{D+1}]

related to the original coordinates by x_i=X_i/X_0. In the new variables, every quadric is defined by an equation of the form

Q(X)=\sum_{ij} a_{ij}X_iX_j=0\,

where the coefficients aij are symmetric in i and j. Regarding Q(X) = 0 as an equation in projective space exhibits the quadric as a projective algebraic variety. The quadric is said to be non-degenerate if the quadratic form is non-singular; equivalently, if the matrix (aij) is invertible.

In real projective space, by Sylvester's law of inertia, a non-singular quadratic form Q(X) may be put into the normal form

Q(X) = \pm X_0^2 \pm X_1^2 \pm\cdots\pm X_{D+1}^2

by means of a suitable projective transformation (normal forms for singular quadrics can have zeros as well as ±1 as coefficients). For surfaces in space (dimension D = 2) there are exactly three nondegenerate cases:

Q(X) = \begin{cases} X_0^2+X_1^2+X_2^2+X_3^2\\ X_0^2+X_1^2+X_2^2-X_3^2\\ X_0^2+X_1^2-X_2^2-X_3^2 \end{cases}

The first case is the empty set.

The second case generates the ellipsoid, the elliptic paraboloid or the hyperboloid of two sheets, depending on whether the chosen plane at infinity cuts the quadric in the empty set, in a point, or in a nondegenerate conic respectively. These all have positive Gaussian curvature.

The third case generates the hyperbolic paraboloid or the hyperboloid of one sheet, depending on whether the plane at infinity cuts it in two lines, or in a nondegenerate conic respectively. These are doubly ruled surfaces of negative Gaussian curvature.

The degenerate form

X_0^2-X_1^2-X_2^2=0. \,

generates the elliptic cylinder, the parabolic cylinder, the hyperbolic cylinder, or the cone, depending on whether the plane at infinity cuts it in a point, a line, two lines, or a nondegenerate conic respectively. These are singly ruled surfaces of zero Gaussian curvature.

We see that projective transformations don't mix Gaussian curvatures of different sign. This is true for general surfaces. [5]

In complex projective space all of the nondegenerate quadrics become indistinguishable from each other.

Probability and statistics

The elliptical distributions, which generalize the multivariate normal distribution and are used in finance, can be defined in terms of their density functions. When they exist, the density functions f have the structure:

f(x)= k \cdot g((x-\mu)\Sigma^{-1}(x-\mu)^T)

where k is a scale factor, x is an n-dimensional random row vector with median vector \mu (which is also the mean vector if the latter exists), \Sigma is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and g is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.[6] The multivariate normal distribution is the special case in which g(z)=e^{-z/2} for quadratic form z.

Thus the density function is a scalar-to-scalar transformation of a quadric expression. Moreover, the equation for any iso-density surface states that the quadric expression equals some constant specific to that value of the density.

See also

References

  1. ↑ Silvio Levy Quadrics in "Geometry Formulas and Facts", excerpted from 30th Edition of CRC Standard Mathematical Tables and Formulas, CRC Press, from The Geometry Center at University of Minnesota
  2. ↑ Sameen Ahmed Khan,Quadratic Surfaces in Science and Engineering, Bulletin of the IAPT, 2(11), 327–330 (November 2010). (Publication of the Indian Association of Physics Teachers).
    Sameen Ahmed Khan, Coordinate Geometric Generalization of the Spherometer and Cylindrometer, arXiv:1311.3602
  3. ↑ Stewart Venit and Wayne Bishop, Elementary Linear Algebra (fourth edition), International Thompson Publishing, 1996.
  4. ↑ Бодренко, И.И. Обобщенные поверхности Дарбу в пространствах постоянной кривизны. Saarbrücken, Germany: LAP LAMBERT Academic Publishing, 2013. C. 119–130. ISBN 978-3-659-38863-7
  5. ↑ S. Lazebnik and J. Ponce, "The Local Projective Shape of Smooth Surfaces and Their Outlines" (PDF)., Proposition 1
  6. ↑ Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275-286.

External links