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PRINCIPAL PLANES AND EIGENVECTORS

Principal plane of a quadric surface. A principal plane of a quadric surface is a plane of symmetry of the surface. For example, in Figure 1 the principal planes of the ellipsoid are the X-Y plane, the X-Z plane and the Y-Z plane. We now give a technical definition of a principal plane.

Def. Principal Plane. A principal plane is a diametral plane that is perpendicular to the chords it bisects.

The equation of the diametral plane conjugate to a system of parallel chords cutting a quadric surface                 

1)        f(x, y, z) = ax² + by² + cz² + 2fyz + 2gxz + 2hxy + 2px + 2qy + 2rz + d = 0

in the direction (l, m, n) is:

2) (al + hm + gn)x + (hl + bm + fn)y + (gl + fm + cn) z + (pl + qm + rn) = 0 .

We recall that a set of direction numbers for the normal to a plane

            ax + by + cz + d = 0

are the coefficients a, b, c. For the condition that the diametral plane be perpendicular to the chords it bisects to be met, the direction of the chords must be the same as the direction the diametral plane’s normal. This means that the coefficients of equation 2) must be proportional to the direction numbers l, m, n, that is that there is a real number k such that

            al + hm + gn = kl

3)        hl + bm + fn = km

            gl + fm + cn = kn

or in matrix form

We recognize this as an eigenvector problem where equation 4) corresponds to as the eigenvector equation AX = λX. We also recognize the matrix e

Let

Then equation 4) becomes

5)        ex = kx

Theorem. A diametral plane is a principal plane if and only if the chords it bisects have direction numbers satisfying the equation ex = kx. Said differently, a diametral plane is a principal plane if and only if the chords it bisects have the direction of an eigenvector of the matrix

Thus the principal planes of a quadric surface are those diametral planes that are perpendicular to the eigenvectors of the matrix e. The eigenvectors of matrix e thus give the directions of the canonical coordinate system axes.

Eigenvectors and principal planes. When one computes the eigenvectors of a quadric surface there will be an eigenvector corresponding to each principal plane (an eigenvector that is perpendicular to it). If there are three principal planes, as in an ellipsoid with three unequal axes, there will be three eigenvectors. If we are dealing with a surface of revolution any plane through the axis of revolution is a principal plane and corresponding to each of these principal planes is an eigenvector. Thus there is a linear manifold of eigenvectors radiating out from the axis of

rotation and perpendicular to it. See Figure 2. If we are dealing with a sphere any plane passing through the center is a principal plane and perpendicular to each principal plane is a corresponding eigenvector. Thus in this case the eigenvectors span all of 3-space, emanating out in every direction from the center (i.e. from the origin of the canonical system).

Equations of the principal planes. If the quadric surface has a center (x₀, y₀, z₀) the equation of a principal plane is given by

            a(x - x₀) + b(y - y₀) + c(z - z₀) = 0

where (a, b, c) are direction numbers for the normal to the plane (i.e. the eigenvector). If the surface is one that has no center then its principal plane can be computed from

            (al + hm + gn)x + (hl + bm + fn)y + (gl + fm + cn) z + (pl + qm + rn) = 0

where (l, m, n) are direction numbers of the normal to the plane (i.e. the eigenvector).

Principal planes of the 17 quadric surfaces. For the seventeen quadric surfaces with equations in canonical form, assuming that no two of the numbers a, b, and c are equal, the principal planes are as follows:

            Surface                      Principal planes

   ------------------------------------------------------------ 

            1 - 6                            x = 0, y = 0, z = 0

            7 - 13                          x = 0, y = 0

            14 - 17                        x = 0

For a quadric cylinder no plane perpendicular to the cylinder axis is a principal plane.

Theorems.

1] Principal planes correspond to non-zero characteristic roots of the matrix e. The number of non-zero characteristic roots of e is equal to the rank of e.

2] Two non-zero characteristic roots of e are equal if and only if the quadric is a surface of revolution.

2] Every quadric surface has a principal plane.