Quadratic forms, canonical forms of quadric surfaces with centers

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QUADRATIC FORMS, CANONICAL FORMS OF QUADRIC SURFACES WITH CENTERS

Quadratic form. A quadratic form is a homogeneous polynomial of degree two. The following are quadratic forms in one, two and three variables:

F(x) = ax²

F(x,y) = ax² + by² + cxy

F(x,y,z) = ax² + by² + cz² + dxy + exz + fyz

The polynomial consists of squared terms for each of the variables plus cross-products terms for all combinations of the variables.

Quadratic forms occur in many branches of mathematics and its applications. They are encountered in the theory of numbers, in crystallography, in the study of surfaces in analytic geometry, and in various problems of physics and mechanics.

It is customary to represent the quadratic form as a symmetric bilinear form

= X^TAX

where A is a symmetric matrix of the coefficients. Note that in this representation each cross-product term appears twice. For example, the x₁x₂term appears as both an x₁x₂term and a x₂x₁term. Because we want the matrix to be symmetric we allocate half of the value of the coefficient to each term. The motivation behind representing the quadratic form in this manner is presumably to facilitate matrix analysis and treatment.

The matrix A is called the matrix of the quadratic form and the rank of A is called the rank of the form. If the rank is less than n the quadratic form is called singular. If the rank is equal to n it is nonsingular.

Application in solid analytic geometry. In solid analytic geometry we are interested in the quadratic form

1) F(x, y, z) = ax² + by² + cz² + 2fyz + 2gxz + 2hxy

which can be written as

or in matrix form as

Note that a quadratic form cannot represent a paraboloid or parabolic cylinder because these surfaces have first degree terms that cannot be eliminated and a quadratic form has no first degree terms. Thus quadratic forms represent only those quadric surfaces which have centers.

Q. What is the relationship between the function

4) f(x, y, z) = ax² + by² + cz² + 2fyz + 2gxz + 2hxy + 2px + 2qy + 2rz + d

and the quadratic form

5) F(x, y, z) = ax² + by² + cz² + 2fyz + 2gxz + 2hxy ?

A. In answer let us note the following two facts:

Fact 1. The same rotation of coordinate system that eliminates the mixed terms in 5) also eliminates the mixed terms in 4). The mixed terms in 4) can be eliminated by a rotation of coordinate system either before the x, y and z terms are removed through a translation or afterwards, it doesn’t matter. Either way, the rotation required to remove the mixed terms is the same. If we know the rotation of coordinate system that eliminates the mixed terms in 5), we know the rotation that eliminates the mixed terms in 4).

Fact 2. Let 4) correspond to a quadric surface with a center. If we translate the x-y-z coordinate system of 4) to a center (x₀, y₀, z₀) we will eliminate the x, y, and z terms of f(x, y, z) and the function will become

6) g(x', y', z') = ax'² + by'² + cz'² + 2fy'z' + 2gx'z' + 2hx'y' + d'

where

d' = px₀ + qy₀ + rz₀ + d

Thus on translation of the coordinate system to the quadric surface center

g(x', y', z') = F(x', y', z') + d'

(where we have simply replaced the variable names x, y, z in the function F(x, y, z) with x', y', z'). Viewing the two functions g(x', y', z') and F(x', y', z') as scalar point functions in 3-space this means they are the same functions throughout the space except for a constant difference d' i.e. they differ in value at each point in space by the amount d'. If we do a suitable rotation on F(x', y', z') we can eliminate the cross product terms and reduce it to canonical form. The same rotation will eliminate the cross product terms in g(x', y', z').

A theorem of matrix theory states the following:

Theorem 1. Every real quadratic form q = X^TAX with symmetric matrix A can be reduced by an orthogonal transformation X = BY to a canonical form

λ₁y₁² + λ₂y₂² + ... + λ_ny_n² ,

where λ₁, λ₂, ... ,λ_n are the characteristic roots of A.

We can restate this theorem for our particular case as follows:

Theorem 2. By a suitable rotation of the coordinate system the quadric surface

F(x, y, z) = ax² + by² + cz² + 2fyz + 2gxz + 2hxy

is reducible to the form

λ₁x² + λ₂y² + λ_nz²

where λ₁, λ₂, λ₃ are the characteristic roots of the matrix

What does this theorem mean for us? This theorem gives us the canonical form of those quadric surfaces that have centers.

Canonical form of quadric surfaces with centers. The canonical form of a quadric with a center depends on whether or not it has a constant term. Nine quadrics have constant terms and the others don’t. The ones with a constant term are:

● real and imaginary ellipsoids

● hyperboloids of one and two sheets

● real and imaginary elliptic cylinders

● hyperbolic cylinder

● real and imaginary parallel planes

Let λ₁, λ₂, λ₃ be the characteristic roots of matrix e and let (x₀, y₀, z₀) be a center. The canonical form of those quadrics with constant terms is

7) λ₁x² + λ₂y² + λ_nz² + d' = 0

where

d' = px₀ + qy₀ + r z₀ + d

The canonical form of those quadrics without constant terms is

8) λ₁x² + λ₂y² + λ_nz² = 0