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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) λ_{1}x + μ_{1}y + ν_{1}z - x_{0} = 0

x_{c}-z_{c} plane (i.e. y_{c} = 0 plane).

2) λ_{2}x + μ_{2}y + ν_{2}z - y_{0} = 0

x_{c}-y_{c} plane (i.e. z_{c} = 0 plane).

3) λ_{3}x + μ_{3}y + ν_{3}z - z_{0} = 0

where λ_{1}, μ_{1}, ν_{1} are the direction
cosines of the x_{c} axis, λ_{2}, μ_{2}, ν_{2} are the
direction cosines of the y_{c} axis, λ_{3}, μ_{3},
ν_{3} are the direction cosines of the z_{c}
axis (with respect to the x-y-z system),
and x_{0}, y_{0} and z_{0} 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} = λ_{1}x + μ_{1}y + ν_{1}z - x_{0}

5) y_{c} = λ_{2}x + μ_{2}y + ν_{2}z - y_{0}

6) z_{c} = λ_{3}x + μ_{3}y + ν_{3}z - z_{0}

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) k_{1}x_{c}^{2} + k_{2}y_{c}^{2} + k_{3}z_{c}^{2} + d_{1} = 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 (x_{0}, y_{0}, z_{0}). 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.

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