本篇論文主要可分作兩大目標，NUOF參數曲線與曲面間交集的求解，以及NUOF級數的頻率最佳化。
NUOF參數曲線與曲面間交集的求解：
NUOF曲線與曲面皆是架構於NUOF級數之上，因其基礎函數是由正餘弦函數所組成，具有連續與無限可微的優點，且NUOF可以作為CAD/CAM與工具機的共同模型，不需要中界的檔案格式。在CAD系統當中，常需要計算圖形之間的交集，而一般常用的可分為曲線與曲線，曲線與曲面以及兩曲面間的交線計算，而採用區間牛頓法則可以在給定的參數範圍內一次求出所有的解。
NUOF級數的頻率最佳化：
NUOF級數可分係數參數與頻率參數，其中頻率部分是屬於非線性變數，在給定一筆資料點的情況之下，將頻率最佳化之後的級數，將可避免掉係數參數值過大以及相鄰資料點間劇烈震盪的情形。

There are two main goals included in this paper : Finding the intersections between the NUOF parametric curve and surface, and the optimalization of the frequency parameters of NUOF series.
Finding the intersections between the NUOF parametric curve and surface
NUOF curve and surface are based on NUOF series. For the reason that the base functions are periodic tirangular functions, they have the advantages of continuity and infinite differentiation.The NUOF can also be used as a common model between CAD/CAM and CNC machine without middle file format.In CAD system, we usually need to find the intersections between graphics.Generally speaking, there are three kinds:two curves, curve and surface, two surfaces.By using Interval Analysis, we can get all solutions in the specified range of parameters.
The optimalization of the frequency parameters of NUOF series
We can separate the NUOF parameters as coefficients and frequency parameters.The frequency parameters are belong to non-linear variables.Given some data points, we have to find the optimal frequency to construct the NUOF series.When we do this, we can avoid the bad coefficient parameters and unexpected vibration between two neighboring data points.

第一章 導論1
1-1研究方向與動機1
1-2 文獻回顧3
1-3 各章摘要6
第二章 區間方法8
2-1前言8
2-2區間數值的意義9
2-3區間數值的演算法則10
2-4區間函式11
2-5利用區間分析解一元非線性方程式13
2-6利用區間分析解多元非線性聯立方程式15
第三章NUOF曲線與曲線交點的求解20
3-1 前言20
3-2 NUOF曲線的性質21
3-3 交點求解的計算法則23
3-4 交點求解程式架構27
3-5 實例測試31
3-6 討論33
第四章 NUOF曲線與曲面交點的求解35
4-1 前言35
4-2 NUOF曲面的性質35
4-3 交集求解的計算法則39
4-4 交集求解程式架構43
4-5 實例測試45
4-6 討論46
第五章 NUOF曲面與曲面交線的求解47
5-1 前言47
5-2 尋找起始點48
5-3 追蹤交線點52
5-4 串接交線點60
5-5 實例測試63
5-6 討論68
第六章 NUOF參數最佳化之討論69
6-1 前言69
6-2 NUOF初始頻率參數的選定70
6-3 NUOF參數最佳化的方法76
6-4 討論82
第七章 NUOF3D程式架構說明83
7-1 實驗軟硬體說明83
7-2 程式核心架構83
7-3 程式介面說明85
第八章 回顧與展望90
8-1 論文回顧90
8-2 未來展望91

[1] Mortenson, M.E., “Geometric Modeling”, John Wiley & Sons , Inc. 1985
[2] Hansen, E.R., “Global Optimization Using Interval Analysis”, Marcel Dekker, Inc. 1992
[3] Moore, R.E., “Interval Analysis”, Presentice-Hall, 1966
[4] Ratschek, H., and Roken, J., “Computer Methods for the Range of Functions”, John Wiley & Sons, 1984
[5] Ratschek, H., and Roken, J., “New Computer Methods for Global Optimization”, John Wiley & Sons, 1988
[6] Vanderplaats, Garret N., “Numerical Optimization Techniques for Engineering Design with Application”, McGraw-Hill International Editions, 1993
[1] 張生福,”NUOF參數化曲線與即時路徑控制”, 國立台灣大學機械工程研究所碩士論文,中華民國八十六年六月
[8] 陳炳德,”使用非線性最佳化頻率曲面做路徑規劃之多軸運動控制器設計”, 國立台灣大學機械工程研究所碩士論文,中華民國八十七年六月
[9] Christoph M. Hoffmann, “Geometric & Solid Modeling : An Introduction”, Morgan Kaufmann Publishers, Inc. 1989
[10] M. Daniel and A. Nicolas, “ A Surface-Surface Intersection Algorithm with a Fast Clipping Technique”, Curve and Surface Geometric Design, pp105-112, 1994
[11] Kriezis, G.A., Prakash, P.V., and Patrikalakis, N.M., “An Method for Intersecting Algebraic Surfaces with Rational Polynomial Patches”, Computer-Aided Design, Vol.22 No. 10, pp.645-654, 1990.
[12] Patrikalakis, N.M., “Surface-to-Surface Intersections”, IEEE CG&A, Vol.13, No.1, pp.89-95, 1993.
[13] Sederberg, T.W., and Meyers, R.J. “Loop Detection in Surface Intersections”, Computer-Aided Geometric Design, Vol.5, pp.161-171, 1988.
[14] Sederberg, T.W., and Christiansen, H.N., and Katz, S., “An Improved Test for Closed Loops in Surface Intersections”, Computer-Aided Design, Vol.21, No.8, pp.505-508, 1989.
[15] 劉志鴻，”B-spline 曲線/曲面交集問題之探討”，國立台灣大學碩士論文，中華民國79年六月
[16] 郭錦誠，”參數化曲線與曲面間交集的探討”，國立台灣大學碩士論文，中華民國84年六月
[17] Neumaier, A., “Interval Methods for Systems of Equations”, Cambridge university Press, 1990
[18] Lane, J.M. and Riesenfled, R., “A Theoretical Development for the Computer Generation and Display of Piece Polynomial Surface”, IEEE Trans. RAMI, Vol.2, pp.35-46, 1980
[19] Koparkar, P.A., and Muder, S.P., “A New Calss of Algorithms for the Processing of Parametric Curves”, Computer-Aided Design, Vol.15 No.1, pp.41-45, Jan.1983