近軸光學是幾何光學中研究「軸對稱光學系統」近軸區成像的一個分支，若系統是理想光學系統時，對近軸區和非近軸區都同樣適用。通常遇到的系統雖然都不是真正的理想，但在光學設計過程中﹐各種像差都得到某種程度的校正，因此近軸光學的計算結果(像的大小、成像位置等)對非近軸區也很正確。因此，近軸光學雖然只描述近軸區的成像性質，但在衡量非近軸區的成像狀況和質量方面仍是不可少的。特別是在光學系統初步設計階段，近軸光學的理論和有關計算公式有其重要的實用意義。本文首先敘述近軸光線之6*6矩陣追蹤法，此方法可追蹤「非軸對稱光學系統」的近軸光線。接著使用近軸光線矩陣追蹤法，推導「軸對稱光學系統」的公式。一般教科書總是有很多令人眼花撩亂的公式，本文利用近軸光線追蹤法，使光學設計者只須將矩陣正確的排入，便可推導前後焦距、前後有效焦距、基點地距離。本文最後則探討稜鏡的像姿改變，並且用透鏡組為例，利用解析解來探討像姿的變化，並將所得的結果呈現出來。 關鍵字:光線追蹤法;像;像姿。

Paraxial optic, a branch of geometric optics, is a way to study axis symmetric optical system in paraxial area. If the system is the ideal optical system, the paraxial area and the non-paraxial area are applicable. Although the system is not ideal commonly, but the various aberrations have been corrected in the optical design process. In that way, the calculated results (such as the size, image location, etc.) by using paraxial optical are also correct in the non-paraxial optical area. Therefore, the paraxial optic just describes the forming image in the paraxial area, but it still does work in the non-paraxial optical area. Especially in the preliminary design stage of the optical system, the theory and formulas of the paraxial optics have important practical significance. In the beginning, we introduce the paraxial matrix ray tracing method; this method can track the paraxial rays in non-axis symmetric optical system. Then we use the ray tracing method to derive formulas of axis symmetric optical system. There are so many dazzling formulas in the textbooks, but we use the paraxial ray tracing method to derive the front and back focal length, effective front and back focal length and the node points. Finally, we discuss the image orientation change in prisms and use the analytical solutions to get the result.

References
1. M. Born, and Wolf, E., Principles of Optics, Pergamon, New York, 1985.
2. R. S. Longhurst, “Geometrical and Physical Optics, Longmans” Green, 1, pp. 42-44, 1964.
3. W.J. Smith, Modern Optical Engineering (3rd ed., Edmund Industrial Optics, Barrington, N.J., 2001
4. K. Lau, R. Hocken, and W. Haight, “ Automatic Laser Tracking Interferometer System for Robot Metrology,” Precision Enginneering 8, 1, pp., 3-8 (1986).
5. F. A. Jenkins and H. E. White, Fundamentals of Optics (4rd ed., WcGraw-Hill, 1981).
6. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1985).
7. P.D. Lin and C. H. Lu, “Analysis and Design of Optical System by Using Sensitivity Analysis of Skew Ray Tracing,” Appl. Opt. 43, 796-807 (2004).
8. P.D. Lin and K. F. Ehmann, “Sensing of Motion Related Errors in Multi-axis Machines,” ASME Journal of Dynamic Systems, Measurement, and Control 118, No. 3, pp. 425-433 (1996).
9. A Nussbaum and R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1976)
10. W. Brouwer, Matrix Method in Optical Instrument Design (Benjamin New York, 1964)
11. J. R. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice-Hall, Englewood Cliffs, N.J., 1972)
12. G. W. Fowles, Introduction too Modern Optics (Holt, Rinehat & Winston, New York, 1975)
13. R. P. Paul, Robot ManipulatorsMathematics (Programming and Control, MIT Press, Cambridge, Mass., 1982).
14. Psang Dain Lin*, Chi-Kuen Sung “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis” Optik 117 (2006) 329–340
15. T. T. Liao and P. D. Lin, “ Analysis of Optical Elements with Flat Boundary Surfaces,” Appl. Opt. 42, 1191-1202 (2003)
16. 蔡忠佑“棱镜成像位姿變化之分析與設計 Prism Analysis and Design Base on Image Orientation Change”成功大學機械系博士論文，中華民國九十六年七月。
17. 薛鈞哲“近軸光線矩陣追蹤法於三維光學系統之分析Analysis and Design of 3-D Optical Systems by Using Matrix-Based Paraxial Ray Tracing Method”成功大學機械系碩士論文，中華民國九十六年六月。析與設計
18. R. A. Hicks and R. K. Perline, "Equiresolution Catadioptric Sensors," Appl. Opt., 44 (29) (2005) pp. 6108-1114.
19. Taylor C. J. and Kriegman D. J. "Vision-Based Motion Planning and Explortion Algorithms for Mobile Rbots," IEEE Trans. on Robotics and Automation, 14 (3) (1998) pp.417-426.
20. Murray, D. and Basu, A., "Motion Tracking with an Active Camera," IEEE Transactions on Pattern Analysis and Machine Intelligence, 16 (5) (1994) pp. 449-459.
21. Taylor C. J. and Kriegman D. J., "Vision-Based Motion Planning and Exploration Algorithms for Mobile Robots," IEEE Trans. on Robotics and Automation, 14 (3) (1998) pp. 417-426.
22. Abdel-Aziz, Y.I. and Karara, H.M., "Direct Linear Transformation from Comparator Coordinates into Object Space Coordinates in Close-Range Photogrammetry," Proceedings of the Symposium on Close-Range Photogrammetry (pp. 1-18), Falls Church, VA: American Society of Photogrammetry, 1971.
23. Hatze, H., "High-Precision Three-Dimensional Photogrammetric Calibration and Object Space Reconstruction Using a Modified DLT-Approach," J. Biomech 21 (1988) pp.533-538.
24. Lenz, R. and Tsai, R., Sep., "Techniques for Calibration of the Scale Factor and Image Center for High Accuracy 3-D Machine Vision Metrology," IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI--10(5) (1988) pp. 713-720.
25. Weng, J. , Cohen, P. and Herniou, M., " Camera Calibration with Distortion models and accuracy evaluation," IEEE Trans. Pattern Analysis and Machine Intelligence, 14, (10) (1992) pp.965-980.
26. Z. Zhang, "A Flexible New Technique for Camera Calibration," IEEE Transactions on Pattern Analysis and Machine Intelligence, 22 (11) (2000) pp.1330-1334.
27. Ma, L. Chen, Y. Q., and Morre, K. L., " Flexible Camera Calibration Using a New Analytical Radial Undistortion Formula with Application to Mobile Robot Localization," Proceedings of the IEEE International Symposium on Intelligence Control, (2003) pp.799-804.
28. Wang, F. Y., "A Simple and Analytical Procedure for Calibrating extrinsic Camewra Parameters," IEEE Trans. Robotics and Automation, 20, (1) (2004) pp.121-124.
29. Wang, F. Y., "An Effecient Coordinate Frame Calibration Method for 3-D Measurement by Multiple Camera Systems," IEEE Trans. Systems, Man, and Cybernetics: Part C, 35 (5) (2005).
30. Klette, R., SchlUns, K. and Koschan, A., Computer Vision: Three-Dimension Data from Images, Springer-Verlag Singapore, Pte. Ltd., p.44, 1998.
31. Foley, J. D., Dam, A. V., Feiner, S. K., and Hughes, J. F., Computer Graphics, Principles and Practices, 2nd Edition, Addision-Wesley Publishing Company, 1981.
32. Wu, C. H., "Robot Accuracy Analysis Based on Kinematics," IEEE Journal of Robotics and Automation, Vol. RA-2/3, (1986) pp. 171-179.
33. Denavit, J. and Hartenberg, R. S., "A Kinematic Notation for Lower Pair Mechanisms Based on Matrices," Transactions of the ASME Journal of applied mechanics, Vol. 77 (1955) pp. 215-221.
34. Uicker, J. J., "On the Dynamic Analysis of Spatial Linkages Using 4x4 Matrices," Dissertation for Doctor of Philosophy, Northwestern University, Evanston, IL., Aug., 1965.
35. Paul, R. P., Robot Manipulators-Mathematics, Programming and Control,
MIT press, Cambridge, Mass., 1982.