逆向工程為從實物、模型或樣品原型中，重建產品CAD模型或產品複製的一種技術。目前在逆向工程技術上仍存在一些問題，同時由於工業上有許多產品的幾何造型十分複雜，使得逆向工程技術在實際上的應用上受到了限制。本研究結合三次元量測、逆向工程曲面重建及CAD/CAM系統等技術，以一套完整的逆向工程CAD模型重建方法，建構出完整的CAD模型。研究中利用曲面量測路徑規劃、量測資料座標重整等方法，以獲得正確的工件模型之量測資料。利用多種的曲面嵌合法可將量測資料嵌合成各種不同形態的曲面模型，多組曲面邊界連續拘束嵌合法則提供適當的邊界連續狀況，可使多組曲面綴片間達到連續且平滑接合的曲面模型，並經由IGES傳輸到商用CAD/CAM系統中與二次曲面整合，重建出一個完整的CAD模型。
曲面嵌合的目的為從一組量測資料中找到最匹配的曲面模型，最佳化之嵌合演算乃是使量測資料至嵌合曲面之均方根誤差值為最小，而得到最佳嵌合曲面。一般產品通常由好幾種曲面形態所組成，每一種曲面形態都有其不同的幾何特性，完整的逆向工程之曲面重建技術需要能夠從量測資料中，重建出適當形態之曲面模型。除了二次曲面、B-spline曲面與舉升曲面等嵌合法外，本研究提出CAD系統常建立的一些曲面形態，如旋轉曲面、掃描曲面、規則曲面、雙線性曲面以及線性昆式曲面等之曲面嵌合法，使重建的曲面模型兼顧工業界複雜幾何形狀的需求與保有原始設計曲面形態的幾何特性。
此外，保有原始設計形狀的連續特性為逆向工程曲面重建技術中另一個相當重要的課題，當鄰近有兩組以上的量測資料時，兩曲面綴片之間的連續性則顯得十分重要。大多數的曲面嵌合法僅能處理單一曲面綴片，而一般產品在兩曲面之間會達到某種程度的連續性，傳統上乃以blending、lofting與stitching等方法使曲面接合，但在接合處無法控制曲面的正確性。本研究提出多組曲面邊界連續的拘束嵌合演算法，可使嵌合曲面在邊界達到G2連續的平滑接合。在嵌合演算過程中，同時嵌合多組量測資料，並利用曲面邊界G0、G1或G2連續等適當的拘束條件，可將每組量測資料嵌合出最佳化的B-spline曲面模型，並使各個曲面綴片之間達到連續且平滑的接合，以保有原始設計曲面與曲面間連續性之特性。
最後，將完整的逆向工程之CAD模型重建方法應用在葉片工件的CAD模型重建，從初始的葉片工件量測至在CAD/CAM系統上建立完整且平滑的葉片工件CAD模型之流程，說明所發展的CAD模型重建技術之適用性。

Reverse engineering is a technology to reconstruct CAD models of existing objects, models or prototypes. Many industrial components or products are complex in geometry. But, reverse engineering has many problems that limit its applications practically. This work focused on the development of an integrated approach, combining three-dimensional digitization, reverse engineering and CAD/CAM technology, for the reconstruction of complete CAD models. A path-planning algorithm and a registration algorithm were developed for part digitization from multiple views. Several surface fitting algorithms were provided to fit the measurement data into appropriate types of surface patches. A constrained surface fitting algorithm was provided also to ensure appropriate connection among multiple surfaces. All surface patches, saved in IGES data format, were integrated with quadratic surface models in a CAD system.
The goal of surface fitting is to provide a surface model that best matches each set of measurement data. The optimization algorithm is typically formulated as a least-squares minimization problem where the root-mean-squares error between the measurement data and the fitted surface is minimized. In surface reconstruction, each set of the measurement data is fitted into an appropriate type of surface patch. In addition to quadratic surface fitting, B-spline surface fitting and lofted surface fitting, several surface fitting algorithms were also provided in this work to yield appropriate geometric model for the data from different topology. Surface fitting algorithms provided are as follows: revolving surface fitting, swept surface fitting, bilinear surface fitting, ruled surface fitting and linear Coons surface etc.
Continuity between the fitted patches for multiple sets of data is one of the critical problems in reverse engineering. When more than two sets of data are adjacent to each other, the continuity between the fitted surfaces becomes an important issue. However, most algorithms available are suited for an isolated surface patch only. A part typically contains multiple surface regions that must be blended to a specified degree of continuity. Conventional approach based on blending, lofting, stitching, etc. cannot guarantee the quality of the surface near the connection region. A constrained surface-fitting algorithm for multiple sets of data was proposed in this study, emphasizing on G2 continuity across the boundary of the fitted surfaces. The proposed surface-fitting algorithm essentially fits several sets of data simultaneously and yields a B-spline patch for each set of data. The G0, G1 and G2 continuity conditions between B-spline surface patches were addressed. Based on these results, additional constraints were specified to achieve G2 continuity across the surface boundary.
Finally, an integrated approach for the reconstruction of complete CAD models was applied to reconstruct the CAD models for the blades. The procedure include measurement path planning, CMM measurement, surface fitting, and reconstructing the complete CAD models of blades in a commercial CAD/CAM system. It is used to demonstrate the feasibility of the proposed approach for practical use.

封面
中文摘要
英文摘要
誌謝
目錄
圖目錄
表目錄
第壹章 緒論
1-1 前言
1-2 文獻回顧
1-3 研究目的與方法
1-4 三次元量測技術
1-4-1 曲面量測路徑規劃
1-4-2 座標重整方法
1-5 論文架構
第貳章 旋轉曲面之嵌合技術研究
2-1 前言
2-2 旋轉曲面模式
2-3 旋轉曲面嵌合演算法
2-3-1 旋轉軸與參數值初始化
2-3-2 曲線控制點座標計算
2-3-3 參數最佳化
2-3-4 旋轉軸最佳化
2-3-5 曲線控制點數目最佳化
2-4 電腦模擬
2-5 實際範例
2-6 結論
第參章 掃描曲面之嵌合技術研究
3-1 前言
3-2 掃描曲面模式
3-3 掃描曲面嵌合演算
3-4 IGES轉換界面
3-5 範例說明
3-6 結論
第肆章 掃描曲面嵌合技術之延伸應用
4-1 前言
4-2 掃描曲之簡化模式
4-3 曲面G連續接合模式
4-3-1 相同曲面形態接合模式
4-3-2 不同曲面形態接合模式
4-4 曲面嵌合演算
4-5 範例說明
4-6 結論
第伍章 多組曲面邊界G連續之嵌合技術研究
5-1 前言
5-2 問題說明
5-3 B-spline曲面連續的邊界條件
5-3-1 位置連續
5-3-2 切線平面連續
5-3-3 曲率連續
5-4 邊界G連續的B-spline曲面嵌合
5-5 範例說明
5-6 結論
第陸章 逆向工程技術在葉片CAD模型重建之應用
6-1 前言
6-2 範例一
6-3 範例二
6-4 結論
第柒章 結論與未來展望
7-1 結論
7-2 未來展望
參考文獻
附錄A B-spline曲線轉換為複合spline曲線

1. Varady, T., Martin, R.R. and Cox, J., "Reverse Engineering of Geometric Models-an Introduction", Computer-Aided Design, Vol. 29, No. 4, pp. 255-268 (1997)
2. Bradley, C., Vickers, G.W. and Milroy, M., "Reverse Engineering of Quadratic Surfaces Employing Three-dimensional Laser Scanning", Proc. Instn. Mech. Engrs., Vol. 208, pp. 21-28 (1994).
3. Chivate, P.N. and Jablokow, A.G., "Solid-model generation from measured point data", Computer-Aided Design, Vol. 25, No. 9, pp. 587-600 (1993).
4. Abella, R.J., Daschbach, J.M. and McNichols, R.J., "Reverse Engineering Industrial Applications", Computers Industrial Engineering, Vol. 26, No. 2, pp. 381-385 (1994).
5. Hoschek, J., "Spline Approximation of Offset Curves", Computer Aided Geometric Design, Vol. 5, pp. 33-40 (1988).
6. Rogers, D.F. and Fog, N.G., "Constrained B-spline Curve and Surface Fitting", Computer-Aided Design, Vol. 21, No. 10, pp. 87-96 (1989).
7. Sarkar, B. and Menq, C.H., "Parameter Optimization in Approximation Curves and Surfaces to Measurement data", Computer Aided Geometric Design, Vol. 8, pp. 267-290 (1991).
8. Sarkar, B. and Menq, C.H., "Smooth-Surface Approximation and Reverse Engineering", Computer-Aided Design, Vol. 23, No. 9, pp. 623-628 (1991).
9. Lai, J.Y. and Liu, C.Y., "Reverse Engineering of Composite Sculptured Surfaces", International Journal of Advanced Manufacturing Technology, Vol. 12, pp. 180-189 (1996).
10. Ye, X., Jackson, T.R. and Patrikalakis, N.M., "Geometric design of functional surface", Computer-Aided Design, Vol. 28, No. 9, pp. 741-752 (1996).
11. Park, H. and Kim, K. "Smooth surface approximation to serial cross-sections", Computer-Aided Design, Vol. 28, No. 12, pp. 995-1006 (1996).
12. Lin, C.Y, Liou, C.S. and Lai, J.Y., "A surface-lofting approach for smooth-surface reconstruction from 3D measurement data", Computers in Industry, Vol. 34, pp. 73-85 (1997).
13. Ma, W. and Kruth, J.P., "Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces", Computer-Aided Design, Vol. 27, No. 9, pp. 663-675 (1995).
14. Pfeifle, R. and Seidel, H.P., "Fitting triangular B-spline to functional scatted data", Compute Graphics Forum, Vol. 15, No. 1, pp. 15-23 (1996).
15. Pottmann, H. and Randrup, T., "Rotational and helical surface approximation for reverse engineering", Computing, Vol. 60, pp. 307-322 (1998).
16. Pottmann, H., Lee, I.K. and Randrup, T., "Reconstruction of kinematic surfaces from scattered data", Symposium on Geodesy for Geometrical and Structural Engineering, Eisenstadt/Austria, pp. 483-488, (1998).
17. Pottmann, H., Chen, H.Y. and Lee, I.K., "Approximation by profiles surfaces", The Mathematics of Surfaces VIII (R. Cripps, ed.), Information Geometers, pp. 17-36 (1998).
18. Faux, I.D. and Pratt, M.J., Computational geometry for design and manufacture, Ellis Horwood, Chichester, UK (1979).
19. Farin, G., "Visual C2 cubic spline", Computer-Aided Design, Vol. 14, No. 3, pp. 137-139 (1982).
20. Kahmann, J., "Continuity of curvature between adjacent Bezier patches", Surfaces in Computer Aided Geometric Design, in Barnhill, R E and Boehm, W (Eds.) North-Holland, Nethelands, pp. 65-75 (1983).
21. Sarraga, R.F., "G1 interpolation of generally unrestricted cubic Bezier cures", Computer Aided Geometric Design, Vol. 4 pp. 23-39 (1987).
22. Boehm, W., "Visual continuity", Computer-Aided Design, Vol. 20, No. 6, pp. 307-311 (1988).
23. Liu, D. and Hoschek, J., "GC1 continuity conditions between adjacent rectangular and triangular Bezier surface patches", Computer-Aided Design, Vol. 21, No. 4, pp. 194-200 (1989).
24. Du, W.H. and Schmitt, J.M., "On the G1 continuity of piecewise Bezier surface: a review with new results", Computer-Aided Design, Vol. 22, No. 9, pp. 556-573 (1990).
25. Choi, B.K., Shin, H.Y. and Yoo, W.S., "Visually smooth composite surfaces for an unevenly spaced 3D data array", Computer Aided Geometric Design, Vol. 10 pp. 157-171 (1993).
26. Barsky, B.A., "End conditions and boundary conditions for uniform B-spline curve and surface representations", Computers in Industry, Vol. 3, pp. 17-29 (1982).
27. Shetty, S. and White, P.R., "Curvature-continuous extensions for rational B-spline curves and surfaces", Computer-Aided Design, Vol. 23, No. 7, pp. 484-491 (1991).
28. Milroy, M.J., Bradly, C., Vickers, G.W. and Weir, D.J., "G1 continuity of B-spline surface patches in reverse engineering", Computer-Aided Design, Vol. 27, No. 6, pp. 471-478 (1995).
29. Rossignac, J.R. and G.Requicha, A.A., "Constant radius blending in solid modeling", Computers in Mechanical Engr, July, pp. 65-73 (1984).
30. Hoffmann, C. and Hopcroft, J., "Quadratic blending surfaces", Computer-Aided Design, Vol. 18, No. 6, pp. 301-306 (1986).
31. Rockwood, A.P. and Owen, J.C., "Blending surfaces in solid modeling", Geometric Modeling, in Farin, G.(ed), SIAM, U.S.A., pp. 367-383 (1987).
32. Flip, D.J., "Blending Parametric surfaces", ACM Transactions on Graphics, Vol. 8, No. 3, pp. 164-173 (1989).
33. Bloor, M.I.G. and Wilson, M.J., "Generating Blending Surfaces Using Partial Differential Equations", Computer-Aided Design, Vol. 21, No. 3, pp. 165-171 (1989).
34. Choi, B.K. and Ju, S.Y., "Constant radius blending in surface modeling", Computer-Aided Design, vol. 21, No. 4, pp. 213-220 (1989).
35. Hartmann, E., "Blending of Implicit Surfaces with Functional Splines", Computer-Aided Design, Vol. 22, No. 8, pp. 500-506 (1990).
36. Choi, B.K., "Surface Modeling for CAD/CAM", Advances in Industrial Engineering, Vol. 11, pp. 261-296 (1991).
37. Bedi, S., "Surface design using functional blending", Computer-Aided Design, Vol. 24, No. 9, pp. 505-511 (1992).
38. FilKins, P.C., Tuohy, S.T. and Patrikalakis, N.M., "Computational Methods for Blending Surface Approximation", Engineering with Computer, Vol. 9, pp. 49-62 (1993).
39. Vida, J., Martin, R.R. and Varady, T., "A Survey of Blending Method that Use Parameteric Surfaces", Computer-Aided Design, Vol. 26, No. 5, pp. 341-365 (1994).
40. Holmstrom, L., "Piecewise quadric blending of implicitly defined surfaces", Computer-Aided Design, Vol. 4, No. 3, pp. 171-189 (1997).
41. Partt, M.J. and Geisow, A.D., "Surface/surface intersection problems", The Mathematics of Surfaces, J.A. Gregory(ed.), Oxford Univ. Press U.K. (1986).
42. Barnhill, R.E., Farin, G., Jordan, M. and Piper, B.R., "Surface/surface intersection", Computer Aided Geometric Design, Vol. 4, No.1, pp. 3-16 (1987).
43. Rasna, D.J. and Ball, T.W., "Procedurally Representing Lofted surfaces", IEEE Computer Graphics & Applications, November, pp. 27-33 (1989).
44. Kriezis, G.A., Prakash, P.V. and Patrikalakis, N.M., "Method for intersection algebraic surfaces with rational polynomial patches", Computer-Aided Design, Vol. 22, No. 10, pp. 645-654 (1990).
45. Hagen, H., etc., "Surface Interrogation Algorithms", IEEE Computer Graphics & Applications, Sep., pp. 53-60 (1992).
46. Barnhill, R.E., Farin, G., Fayard, L. and Hagen, H., "Twists, Curvatures and Surface Interrogation", Computer-Aided Design, Vol. 20, No. 6, pp. 341-346 (1988).
47. 劉奎江, ?-D葉片自動量測規劃及逆向工程之研究", 中央大學機械工程研究所碩士論文 (1998).
48. 姚嘉瑜, "量測點資料處理及CT影像CAD模型重建", 中央大學機械工程研究所碩士論文 (1997).
49. Choi, B.K., Surface Modeling for CAD/CAM Elsevier, New York (1991).
50. Rogers, D.F. and Adams, J.A., Mathematical Elements for Computer Graphics McGRAW-Hill, New York (1990).
51. Zeid, I., CAD/CAM theory and practice, McGraw-Hill, New York (1991).
52. Hartley, P.J., and Judd, C.J., "Parametrization and Shape of B-spline Curves for CAD", Computer-Aided Design, Vol. 12, No. 5, pp. 235-238 (1980).
53. Reklaitis, G.V., Ravindran, A. and Ragsdell, K.M., Engineering Optimization Method and Applications, John Wiley and Sons, (1984).
54. Bartels, R.H., Beatty, J.C. and Barsky, B.A., An Introduction of Splines for use in Computer Graphics and Geometric Modeling, Morgan Kaufmann Publishers, Inc., California, (1987).
55. Choi, B. K. and Lee, C. S., "Sweep Surface Modeling via Coordinate Transformations and Blending", Computer-Aided Design, Vol. 22, No. 2, pp. 87-96 (1990).
56. U.S. National Institute of Standards and technology, Initial Graphics Exchange Specification (IGES) Version 5.0 NISTIR 4412 (1990).