目前使用的光學系統大多為球面邊界，但是會因為像差而產生成像缺陷，導致系統品質下降。非球面邊界可用於改善球面邊界的缺點，由於製作上的困難，使得價格上也較為昂貴，但在光學領域中仍具有相當的重要性，其中由環曲面邊界組成之光學元件，即為非球面元件中值得深入探討與研究的對象。

本文藉由歪斜光線追蹤法求得環曲面的結構參數，由光線的入射點與折（反）射光線方向，推導出光線經過環曲面邊界的成像方程式，並利用相關參數對元件作出“光源變化”與“元件位姿偏移量”所造成的靈敏度分析，並利用Matlab電腦程式進行分析，以期能使環曲面光學元件能有更佳的成像品質。

文末提到環曲面透鏡於矯正眼睛散光的應用，由於光線通過散光患者的眼睛時，各方向的光線聚焦情況皆不相同，而環曲面透鏡也具有相同特性，因此可藉由環曲面透鏡中和因散光造成聚焦不完美的問題，最後根據五種散光案例作矯正的模擬。

The spherical components were the most popular parts for optical systems used today, because they are easy to manufacture and low in cost. But the aberration cause by the light propagate through a spherical boundary is inevitable and may degrade the quality of image. Aspherical boundary surfaces are more difficult and expensive to manufacture. However, there are some cases where components with aspherical boundary surfaces have a significant advantage over spherical ones. Torus is one of the aspherical boundary surfaces which is worthy for studying.
In this thesis, we applied the well-known skew-ray-tracing method to obtain all the parameters of a toric structure and derived the image-forming equation according to the refracted/reflected vectors and the incident points. Than the sensitivity analysis method was obtained by direct mathematical analysis based on differential changes in incident light source or orientation of optical element. We expected the way of better image quality can be achieved by using Matlab to analysis the method we developed.
We also discussed the applications of toric lenses which used in astigmatism. In this investigation, we can know that the parallel light rays go through the toric lenses will focus on two points, so that we can use the characteristic to adjust the human eyes with astigmatism. In the end of this thesis, we proposed simulations for five different types of astigmatism which composed of toric boundary surfaces and spherical boundary surfaces.

1. H. Haber, "Torus grating," J. Opt. Soc. Am. 40, 153-165 (1950)
2. M. R. Descour, D. I. Simon, and W. -H. Yeh, "Ring-Toric Lens for Focus-Error Sensing in Optical Data Storage," Appl. Opt. 38, 1388-1392 (1999)
3. J. J. Zambuto, R. E. Gerber, J. K. Erwin, and M. Mansuripur, "Ring lens focusing and push-pull tracking scheme for optical disk systems," Appl. Opt. 33, 7987- (1994)
4. T. R. Ferguson and M. E. Smithers, "Toric unstable resonators," Appl. Opt. 23, 2122- (1984)
5. E. Hecht, "Optics," 4th edition, Addison Wesley Longman, Inc., New York, 2002
6. W. J. Smith, "Modern Optical Engineering," Third Edition, SPIE Press. McGRAW-HILL, 2000.
7. R. S. Longhurst , "Geometrical and Physical Optics," Longmans Green, pp.42-44, 1964.
8. A. E. Murray, "A toric skew ray trace," J. Opt. Soc. Am. 44, 672- (1954)
9. A. E. Murray, "Skew astigmatism at toric surfaces, with special reference to spectacle lenses," J. Opt. Soc. Am. 47, 599- (1957)
10. J. Barcala, M. C. Vazquez, and A. Garcia, "Optic systems with spherical, cylindrical, and toric surfaces," Appl. Opt. 34, 4900- (1995)
11. P. D. Lin, "Analysis and Modeling of Optical Elements and Systems," Transactions of the ASME Journal of Engineering for Industry, Vol.116, pp.101-107, February 1994.
12. P. D. Lin and T. T Liao, "Skew-Ray Tracing and Sensitivity Analysis of Geometrical Optics," Transactions of the ASME Journal of Manufacturing Science and Engineering, Vol.122, pp. 338-349, 2000.
13. P. D. Lin and C. H. Lu, "Analysis and Design of Optical System by Use of Sensitivity Analysis of Skew Ray Tracing," Applied Optics, Vol.43, pp. 796-807, 2004.
14. P. D. Lin and T. T. Liao, " Analysis of Optical Elements with Flat Boundary Surfaces, " Applied Optics, Vol. 42, No. 7, pp. 1191-1202, 2003.
15. M. Laikin, Lens Design, Second Edition, New York, Marcel Dekker, Inc. 1995.
16. J. D. Foley, A. V. Dam, S. K. Feiner and J. F. Hughes, Computer Graphics, Principles and Practices, 2nd Edition, Addision-Wesley Publishing Company 1981.
17. R. P. Paul, Robot Manipulators-Mathematics,Programming and Control, MIT press, Cambridge, Mass., 1982.
18. C. H. Wu, “Robot Accuracy Analysis Based on Kinematics,” IEEE Journal of Robotics and Automation, Vol. RA-2/3, pp.171-179, 1986.
19. J. Denavit and R. S. Hartenberg, “A Kinematic Notation for Lower Pair Mechanisms Based on Matrices,” Transactions of the ASME Journal of applied mechanics, Vol. 77, pp.215-221, 1955.
20. 12 J. J. Uicker, “On the Dynamic Analysis of Spatial Linkages Using 4x4 Matrices,” Dissertation for Doctor of Philosophy, Northwestern University,Evanston, ILL., Aug. 1965.
21. M. Born and E. Wolf, "Principles of Optics", Pergamon, New York, 1985.
22. 徐俊昇，"光線在拋物面、橢圓面和雙曲面之追蹤及其靈敏度分析"，碩士論文，國立成功大學機械工程研究所，2003年6月。
23. 林慶國，"圓柱體透鏡的光線追蹤"，碩士論文，國立成功大學機械工程研究所，2007年6月。
24. 正中書局編輯委員會，"高等方程式論"，正中書局出版，1965
25. 徐氏基金會科學圖書編譯委員會，"眼鏡光學"，徐氏基金會出版，1977。
26. 徐氏基金會科學圖書編譯委員會，"眼鏡配置學"，徐氏基金會出版，1979。
27. 徐氏基金會科學圖書編譯委員會，"眼鏡片製作實務"，徐氏基金會出版，1979。
28. 徐氏基金會科學圖書編譯委員會，"眼鏡師訓練手冊"，徐氏基金會出版，1983。