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GENERAL EQUATION OF THE SECOND DEGREE, QUADRIC SURFACES, REDUCTION TO CANONICAL FORM, THE 17 CANONICAL FORMS, TRANSFORMATION OF COORDINATES

General equation of the second degree. The general equation of the second degree in three variables is

1) f(x, y, z) = ax^{2} + by^{2} + cz^{2} + 2fyz + 2gxz + 2hxy + 2px + 2qy + 2rz + d = 0.

This equation has as its locus a surface in space called a quadric surface or a conicoid. See figure 1. Quadric surfaces consist of ellipsoids, hyperboloids, paraboloids, and certain limiting or degenerate forms of these.

Simplification by a change of coordinate system. The algebraic expression for any curve or surface is dependent on the location and orientation of the coordinate system. When the coordinate system is rotated or moved in any way the expression for the curve or surface changes. The general equation of the second degree can be simplified greatly by a change to a different coordinate system. A suitable rotation of the coordinate system will eliminate all of the mixed terms in xy, xz and yz. A suitable translation will eliminate most or all of the first degree terms in x, y and z. By a suitable translation and rotation of the coordinate system the general second degree equation can be reduced to one of the following canonical forms:

Let us make some observations concerning some of the surfaces.

● An ellipsoid can be conceived of as a surface obtained from a sphere by uniformly stretching (or compressing) the sphere in three mutually perpendicular directions by given amounts. The equation of a sphere centered at the origin is

2) x^{2} + y^{2} + z^{2} = 1

and the equation of an ellipsoid centered at the origin is

If (x, y, z) is a point satisfying equation 2) then (ax, by, cz) is a point satisfying equation 3). Thus if P (x, y, z) is a point on sphere 2) then P'(ax, by, cz) is the corresponding point on ellipsoid 3).

● The hyperboloid of one sheet (Form 3), hyperboloid of two sheets (Form 4), second-order cone (Form 5), and elliptic paraboloid (Form 7) all appear in special cases as surfaces of revolution. The general forms of these surfaces can be viewed as produced by a uniform stretching of their surface of revolution form in a way similar to the way an ellipsoid can be produced from a sphere by stretching. For example, the equation of the hyperboloid of one sheet is

If a = b it is a surface of revolution with equation

Surface 4) can be obtained from surface 5) by a stretching in the y direction by a coefficient b/a. The same is true of the other surfaces — the hyperboloid of two sheets, second-order cone, etc. We can take the surface in its surface of revolution form and stretch it in a direction perpendicular to its axis, in the x or y direction, to get a general form.

● One can learn a lot about surfaces such as these by examining cross sections. For example, one can examine cross sections perpendicular to the z axis of 5) by setting z = k and see that they are circles. We can also look at a cross section created by a plane passing through (containing) the z axis. For example, if we substitute y = 0 into 5) we get the equation of the intersection of the surface with the plane y = 0. We note that it is a hyperbola. Such curves obtained by passing planes through the axis of revolution are called meridians. A surface of revolution can be obtained by rotating a meridian about the axis of revolution.

● The forms 9 - 17 are all represented by equations in only the two variables x and y (the variable z is missing). They thus all correspond to cylinders whose directrices are the corresponding curves in the x-y plane and whose generators are lines parallel to the z axis.

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TRANSFORMATION OF COORDINATES

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To transform an equation of a surface from an old system of rectangular coordinates (x, y, z) to a new system of rectangular coordinates (x', y', z'), substitute for each old variable in the equation of the surface its expression in terms of the new variables.

Translation of coordinate system. Let the origin of the new x'-y'-z' system be at point (h, k, l) of the old x-y-z system with the axes of the new system parallel to the corresponding axes of the old system. Then

x = x' + h

y = y' + k

z = z' + l

See Figure 1.

Rotation of the axes about the origin. Let the origin of the new x'-y'-z' system be coincident
with the origin of the old system and let λ_{1}, μ_{1}, ν_{1} be the direction cosines of the x' axis, λ_{2}, μ_{2}, ν_{2}
be the direction cosines of the y' axis, λ_{3}, μ_{3}, ν_{3 }be the direction cosines of the z' axis.

Then

x = λ_{1}x' + λ_{2}y' + λ_{3}z'

1) y = μ_{1}x' + μ_{2}y' + μ_{3}z'

z = ν_{1}x' + ν_{2}y' + ν_{3}z'

x' = λ_{1}x + μ_{1}y + ν_{1}z

2) y' = λ_{2}x + μ_{2}y + ν_{2}z

z' = λ_{3}x + μ_{3}y + ν_{3}z

We can write these equations in matrix form as

where the matrices are called rotation matrices. Multiplication by a rotation matrix transforms the coordinates of a point from one system to another system.

Note. Equation 1) is best understood in vector terms as a change of basis where 1) is equivalent to

the vectors

representing orthogonal, unit basis vectors.

Equations relating the coordinates of the
original and canonical coordinate systems of a
quadric surface. Let x, y and z be the coordinates of a
point P with respect to the original X-Y-Z coordinate
system and x_{c}, y_{c} and z_{c} be the coordinates of the point
with respect to the canonical X_{c}-Y_{c}-Z_{c} coordinate system. See Figure 3. Let the origin of the
canonical X_{c}-Y_{c}-Z_{c} system be located at (x_{0}, y_{0}, z_{0}) and let λ_{1}, μ_{1}, ν_{1} be the direction cosines of
the x_{c} axis, λ_{2}, μ_{2}, ν_{2} be the direction cosines of the y_{c} axis, λ_{3}, μ_{3}, ν_{3 }be the direction cosines of
the z_{c} axis with respect to the X-Y-Z coordinate system. Then the relationship between the
coordinates x, y, and z and x_{c}, y_{c} and z_{c} is given by the following equations:

and

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