EQUATIONS RELATING THE CANONICAL SYSTEM COORDINATES TO THE EXPRESSIONS FOR THE COORDINATE PLANES

Equations of the canonical system coordinate planes. The equations of the canonical system coordinate planes, expressed in normal form, are

y_c-z_c plane (i.e. x_c = 0 plane).

1)        λ₁x + μ₁y + ν₁z - x₀ = 0

x_c-z_c plane (i.e. y_c = 0 plane).

2)        λ₂x + μ₂y + ν₂z - y₀ = 0

x_c-y_c plane (i.e. z_c = 0 plane).

3)        λ₃x + μ₃y + ν₃z - z₀ = 0

where λ₁, μ₁, ν₁ are the direction cosines of the x_c axis, λ₂, μ₂, ν₂ are the direction cosines of the y_c axis, λ₃, μ₃, ν₃ are the direction cosines of the z_c axis (with respect to the x-y-z system), and x₀, y₀ and z₀ are the perpendicular distances from the x-y-z system origin to the respective canonical system coordinate planes.

Theorem 1. For any point P (x, y,z) in space the following relationship between the coordinates (x_c, y_c, z_c) and (x, y, z) holds

4)        x_c = λ₁x + μ₁y + ν₁z - x₀ 

5)        y_c = λ₂x + μ₂y + ν₂z - y₀

6)        z_c = λ₃x + μ₃y + ν₃z - z₀

where x_c, y_c, z_c are the coordinates of point P with respect to the x_c- y_c- z_c canonical coordinate system.

Note that the right member of 4) corresponds to the left member of 1), the right member of 5) corresponds to the left member of 2), etc.. We can state this theorem in words as follows: Each of the canonical system coordinates x_c, y_c, and z_c is equal to the expression for its coordinate plane as expressed in normal form.

What does this mean? It provides us with an roundabout method of finding the expressions for x_c, y_c and z_c in terms of x, y and z i.e. the equations 4), 5), and 6) above. After we compute the eigenvectors for a surface we are in most cases able to write the equations for the three coordinate planes. We can then put those equations for the coordinate planes in normal form and then, from them, write the equations 4), 5), and 6) above thus giving x_c, y_c znd z_c in terms of x, y and z.

Suppose we have the canonical equation

7)                    k₁x_c² + k₂y_c² + k₃z_c² + d₁ = 0

for some surface. If we can substitute the expressions for x_c, y_c, and z_c of equations 4), 5), and 6) into equation 7) and expand we should get back our original equation in x, y and z. Doing so provides a check on our work and may be of use for other reasons.

Proof of Theorem 1. If we write equations 4), 5) and 6) in matrix form we get

To show that the assertion of the theorem is true let us first construct an intermediate x_w-y_w-z_w coordinate system which has the same origin as the x-y-z system but rotated so as to have the same orientation as the canonical x_c-y_c-z_c system i.e. its axes are parallel to the corresponding x_c-y_c-z_c system axes. In the intermediate x_w-y_w-z_w coordinate system the coordinates of a point P are given by

In the intermediate x_w-y_w-z_w coordinate system the coordinates of the origin of the canonical x_c-y_c-z_c system are (x₀, y₀, z₀). The relationship between the coordinates of a point P as expressed in the x_w-y_w-z_w system and in the x_c-y_c-z_c system is

Substituting 10) into 9) gives equation 8).

End of proof.