CHANGES IN EQUATION COEFFICIENTS PRODUCED BY TRANSLATION AND ROTATION OF THE COORDINATE SYSTEM

Changes in equation coefficients produced by translation. What changes are produced in the coefficients of the general equation of the second degree

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

by a translation of the coordinate system? Let x'-y'-z' be a coordinate system whose origin is located at point (x₀, y₀, z₀) of an x-y-z system with corresponding axes parallel. Let us observe the result of substituting

x = x' + x₀

y = y' + y₀

z = z' + z₀

into equation 1) above.

The equation becomes

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

            = 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 that a translation has no effect on the coefficients of the second degree terms (i.e. the coefficients a, b, c, f, g, h). It only changes the first degree coefficients p, q, r and the constant term d where

            p ax₀ + hy₀ + gz₀ + p

            q hx₀ + by₀ + fz₀ + q

            r gx₀ + fy₀ + cz₀ + r 

            d f(x₀, y₀, z₀)

Changes in equation coefficients produced by rotation of the coordinate system

about it origin. Let the origin of a x'-y'-z' system be coincident with the origin of an x-y-z system and let λ₁, μ₁, ν₁ be the direction cosines of the x' axis, λ₂, μ₂, ν₂ be the direction cosines of the y' axis, λ₃, μ₃, ν₃ be the direction cosines of the z' axis as referred to the x-y-z system.

 Then

            x = λ₁x' + λ₂y' + λ₃z'

            y = μ₁x' + μ₂y' + μ₃z'

            z = ν₁x' + ν₂y' + ν₃z'

Question. What effect does this rotation of the coordinate system have on the coefficients of the general equation of the second degree 1)? 

Answer. A rotation of the coordinate system causes changes in all the coefficients a, b, c, f, g, h, p, q, r. The constant term d, however, is unchanged.