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Let us explore the question of what translation of the coordinate system might reduce the general equation of the second degree

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

to simpler form. More specifically, let us ask what translation might cause the first degree terms to disappear. Let us observe the result of substituting


            x = x' + x0

            y = y' + y0

            z = z' + z0

into equation 1) above.

The equation becomes

2)        f(x', y', z') = f(x' + x0, y' + y0, z' + z0) = a(x' + x0)2 + b(y' + y0)2 + c(z' + c0)2 + f(y' + y0)(z' + z0) + 2g (x' + x0)(z' + z0) + 2h(x' + x0)(y' + y0) + 2 p(x' + x0) + 2q (y' + y0) + 2r(z' + z0) + d

            = ax' 2 + by' 2 + cz' 2 + 2fy'z' + 2gx'z' + 2hx'y' + 2(ax0 + hy0 + gz0 + p)x' + 2(hx0 + by0 + fz0 + q)y' + 2(gx0 + fy0 + cz0 + r)z' + ax02 + by02 + cz02 + 2f y0z0 + 2gx0z0 + 2hx0y0 + 2px0 + 2qy0 + 2rz0 + d

and, finally

3)        f(x', y', z') = ax' 2 + by' 2 + cz' 2 + 2fy'z' + 2gx'z' + 2hx'y' + 2(ax0 + hy0 + gz0 + p)x' + 2(hx0 + by0 + fz0 + q)y' + 2(gx0 + fy0 + cz0 + r)z + f(x0, y0, z0) = 0

We see from 3) above that all the coefficients a, b, c, f, g, h of the second degree terms remain unchanged under the translation. For the first degree terms to disappear we can see that the coefficients 2(ax0 + hy0 + gz0 + p), 2(hx0 + by0 + fz0 + q), and 2(gx0 + fy0 + cz0 + r) of x', y' and z' must be zero. Thus the first degree terms should be eliminated by a translation to that (x0, y0, z0) satisfied by the following set of equations:

            ax0 + hy0 + gz0 + p = 0

4)        hx0 + by0 + fz0 + q = 0

            gx0 + fy0 + cz0 + r = 0

Let us note that this set is precisely the set of equations defining a center of a quadric surface. Thus we have shown that the quadric surface centers do represent points at which the first degree terms disappear. So if we do a translation to a center the second degree terms remain the same and the first degree terms disappear. Now let us ask what the constant term d becomes on a translation to a center. Let d' denote the constant term at a center. From 3) above we see that d' = f(x0, y0, z0). What is the value of f(x0, y0, z0) at a center? The function

f(x0, y0, z0) = ax02 + by02 + cz02 + 2fy0z0 + 2gx0z0 + 2hx0y0 + 2px0 + 2qy0 + 2rz0 + d

can be written as

f(x0, y0, z0) = (ax0 + hy0 + gz0 + p)x0 + (hx0 + by0 + fz0 + q)y0 + (gx0 + fy0 + cz0 + r)z0 + (px0 + qy0 + rz0 + d)

and we can see from inspection of this equation and 4) above that the first three terms vanish at a center (x0, y0, z0). Thus the new constant term has the value

            d' = f(x0, y0, z0) = px0 + qy0 + rz0 + d

Theorem. If a translation is made to a center (x0, y0, z0) of the quadric surface

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

the transformed equation becomes

            f(x', y', z') = ax' 2 + by' 2 + cz' 2 + 2fy'z' + 2gx'z' + 2hx'y' + px0 + qy0 + rz0 + d = 0

Theorem. The equation obtained from that of a given quadric surface by a translation to a center of the surface is independent of the center chosen.

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