APPENDIX B
EQUATION OF A GENERAL FORM OF TORUS
If the translation da along the z axis of the local
fixed coordinate system is disregarded in Eq. (2.5), (see
Fig. 2.11), we can obtain the following equations:
Cx = C8a(bCGb + a) Sea(bSebCcb sbSab) (B.1)
Cy = S8a(bC8b + a) + C8a(bS9bCab sbSab) (B.2)
Cz = bSebScb + sbCab (B.3)
Squaring and adding Eqs. (B.1) (B.3) yields
Cx2 + y2 + C2 = a2 + b2 + Sb2 + 2abC8b (B.4)
Multiplying Eq. (B.4) by Sab yields
Sab{(Cx2 + Cy2 + Cz2) (a2 + b2 + Sb2)} = 2abCebSab
(B.5)
Multiplying Eq. (B.3) by 2a yields
2a(C, sbCab) = 2abS8bSab (B.6)
Finally, squaring and adding Eqs. (B.5) and (B.6) and then
rearranging terms yields the following equation:
183