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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
yc-zc plane (i.e. xc = 0 plane).
1) λ1x + μ1y + ν1z - x0 = 0
xc-zc plane (i.e. yc = 0 plane).
2) λ2x + μ2y + ν2z - y0 = 0
xc-yc plane (i.e. zc = 0 plane).
3) λ3x + μ3y + ν3z - z0 = 0
where λ1, μ1, ν1 are the direction cosines of the xc axis, λ2, μ2, ν2 are the direction cosines of the yc axis, λ3, μ3, ν3 are the direction cosines of the zc 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 (xc, yc, zc) and (x, y, z) holds
4) xc = λ1x + μ1y + ν1z - x0
5) yc = λ2x + μ2y + ν2z - y0
6) zc = λ3x + μ3y + ν3z - z0
where xc, yc, zc are the coordinates of point P with respect to the xc- yc- zc 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 xc, yc, and zc 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 xc, yc and zc 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 xc, yc znd zc in terms of x, y and z.
Suppose we have the canonical equation
7) k1xc2 + k2yc2 + k3zc2 + d1 = 0
for some surface. If we can substitute the expressions for xc, yc, and zc 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 xw-yw-zw 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 xc-yc-zc system i.e. its axes are parallel to the corresponding xc-yc-zc system axes. In the intermediate xw-yw-zw coordinate system the coordinates of a point P are given by
In the intermediate xw-yw-zw coordinate system the coordinates of the origin of the canonical xc-yc-zc system are (x0, y0, z0). The relationship between the coordinates of a point P as expressed in the xw-yw-zw system and in the xc-yc-zc system is
Substituting 10) into 9) gives equation 8).
End of proof.
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