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The general equation of the second degree

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

represents one of 17 different canonical forms which correspond to 17 canonical surfaces. Any equation of the second degree can be reduced to canonical form by a suitable translation and rotation. Thus when faced with a particular second degree equation the following questions immediately present themselves:

1] Which of the 17 quadric surfaces does this equation represent?

2] What is equation of the surface in the canonical coordinate system?

3] What is the location of the origin of the canonical system?

4] What is the orientation of the canonical coordinate system? In other words, in what directions do the canonical system axes point?

5] What are the equations of the principal planes of the surface?

We shall now deal with these questions. Certain invariants of the quadric surface including the ranks of matrices E and e and the eigenvalues of matrix e allow us to answer the questions without actually doing a translation and rotation of the coordinate system, a circumstance that is very fortunate because doing the actual translation and rotations is generally very laborious. We define the following:

D — value of determinant of matrix e

Δ — value of determinant of matrix E

ρ3 — rank of matrix e

ρ4 — rank of matrix E

k1, k2, k3 ---- eigenvalues (or characteristic roots) of matrix e

\

Computation of the eigenvalues and eigenvectors. We wish to compute the eigenvalues and eigenvectors of matrix e. The eigenvector equation that we wish to solve is

ex = kx

or, more explicitly,

which can also be written as

The characteristic equation is

which is equal to

5)        k3 + Ik2 + Jk - D = 0

where

D is the determinant of e

I = a + b + c

J = ab + ac + bc - f2 - g2 - h2

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Note. I is the sum of the elements of the main diagonal of e and J is the sum of their cofactors. Thus if we denote the cofactors of elements a, b, c of e by A, B, C then the characteristic equation can be written as

k3 + (a + b + c)k2 + (A + B + C)k - D = 0

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The procedure is to solve the characteristic equation 5) for k i.e. to find the roots of the equation.. Each root corresponds to an eigenvalue (or characteristic value). We then plug each eigenvalue back into equation 3) and solve that equation for x where

is an eigenvector. Thus for each eigenvalue we compute an eigenvector.

Eigenvectors are also called characteristic vectors.

Def. Characteristic direction. The direction of an eigenvector given in terms of the direction cosines (λ, μ, ν).

Identifying the surface. We identify a surface using four pieces of information: the values of ρ3 and ρ4, the sign of Δ, and the question of whether the k’s are, or are not, all of the same sign. ρ3 and ρ4 are the ranks of e and E respectively, Δ is the value of | E |, and the k’s are the eigenvalues. We make the identification using the following table:

 Number Surface Sign of k’s same sign? 1 Ellipsoid 3 4 - yes 2 Imaginary ellipsoid 3 4 + yes 3 Hyperboloid of one sheet 3 4 + no 4 Hyperboloid of two sheets 3 4 - no 5 Second-order cone 3 3 no 6 Imaginary second-order cone 3 3 yes 7 Elliptic paraboloid 2 4 - yes 8 Hyperboloic paraboloid 2 4 + no 9 Elliptic cylinder 2 3 yes 10 Imaginary elliptic cylinder 2 3 yes 11 A pair of intersecting imaginary planes 2 2 yes 12 Hyperbolic cylinder 2 3 no 13 A pair of intersecting planes 2 2 no 14 Parabolic cylinder 1 3 15 A pair of parallel planes 1 2 16 A pair of imaginary parallel planes 1 2 17 A pair of coincident planes 1 1

From Olmstead. Solid Analytic Geometry. p. 192

The equation in the canonical coordinate system.

The equation of a quadric surface with a center. The equation 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 equation in the canonical coordinate system of those quadric surfaces with constant terms is

λ1x2 + λ2y2 + λnz2 + d' = 0

where

d' = px0 + qy0 + r z0 + d

The equation of those quadric surfaces without constant terms is

λ1x2 + λ2y2 + λnz2 = 0

The equation of quadric surfaces without centers. The equations for the three quadric surfaces that do not have centers are:

1] Elliptic and hyperbolic paraboloids. The equation is

λ1x2 + λ2y2 + 2r'z = 0

where the value of r' is determined from the relation Δ = - k1k2r'2.

2] Parabolic cylinder. The equation is

x2 + 2r'z = 0

where the method for determining the value of r' is complicated. We won’t go into it.

Determining location of canonical system origin. The location of the origin of the canonical system is at the surface center for those 14 surfaces that have centers. In the other three cases, surfaces 7, 8 and 14 representing the elliptic and hyperbolic paraboloids and the parabolic cylinder, the origin of the canonical system is located at a vertex. Of the surfaces with centers, surfaces 1 - 6 have a single center, surfaces 9 - 13 have lines of centers, and surfaces 15 - 17 have planes of centers. For those surfaces with multiple centers any center may be selected for the canonical system origin. The elliptic and hyperbolic paraboloids have a single vertex and the parabolic cylinder has a line of vertices. In the case of the parabolic cylinder any vertex may be selected for the origin.

Surfaces having a center. For surfaces having centers, the location of the center or centers is determined by solving the following system of equations:

ax + hy + gz + p = 0

hx + by + fz + q = 0

gx + fy + cz + r = 0

or in matrix form

Surfaces without centers. Determining the vertices of the elliptic and hyperbolic paraboloids and the parabolic cylinder is roundabout and best explained by example.

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Example. Olmstead , Solid Analytic Geometry, pp. 196, 197

Surface analysis.

Equation: x2 + 3y2 + z2 + 2xy + 2xz + 2yz - 2x + 4y + 2z + 12 = 0

ρ3 = 2

ρ4 = 4

D = 0

Δ = -8

Characteristic equation: k3 - 5k2 + 4k = 0

Roots of characteristic equation: 0, 1, 4

Surface identification: elliptic paraboloid

Canonical equation: The canonical form of the equation is

λ1x2 + λ2y2 + 2r'z = 0

where the value of r' is determined from the relation Δ = - k1k2r'2.

Thus

Equation of surface:

Directions of eigenvectors.

(1, -1, 1) corresponding to k = 1

(1, 2, 1) corresponding to k = 4

Principal planes.

1)        x - y + z - 2 = 0                     corresponding to k = 1

2)        x + 2y + z + 1 = 0                   corresponding to k = 4

Note. The equation of the principal plane normal to an eigenvector with direction (l, m, n) is

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

Direction numbers of the line of intersection of the two principal planes: (1, 0, -1)

Note. These direction numbers were obtained from the following theorem:

Theorem. A set of direction numbers for the line of intersection of the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is given by

We now substitute into the canonical equation

the expressions for x', y', and z' in terms of x, y, and z. These expressions for x', y', and z' are given by

where the right members of these expressions correspond to the expressions for the principal planes as expressed in normal form [i.e. they correspond to the expressions for the normal forms of the principal planes (or coordinate planes)]. Substituting into the canonical equation we get

which, on expansion gives the original equation. Thus, on expansion, we can determine the value of the unknown variable δ. Expansion shows that the + sign is correct and that δ = -5 and therefore

The point of intersection of the principal planes and the plane x - z - 5 = 0 is the vertex (and the origin that we seek). That point of intersection is (3, -1, -2).

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Determining the orientation of the canonical system. We determine the orientation of the canonical system using eigenvectors that we compute. The computed eigenvectors correspond to the directions of the coordinate axes in the canonical system. In the case of surfaces of revolution there will be a single distinct eigenvector corresponding to the axis of symmetry and plus a manifold of eigenvectors in the plane perpendicular to the axis of symmetry. In this case we use the distinct eigenvector for one axis and then arbitrary select an eigenvector in the manifold for another axis and then another in the manifold perpendicular to it for the third axis.

Equations of the principal planes. The equation of the principal plane normal to an eigenvector with direction (l, m, n) is

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

For surfaces with centers, the principal plane is also given by the equation

lx + my + nz - lx0 - my0 - nz0 = 0

where (x0, y0, z0) is the center.

References.

Olmstead. Solid Analytic Geometry. Chap. 8