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Research on Approximation and Convergence
Problems in Curves and Surfaces Modeling
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2007 � 4 V厦门大学博硕士论文摘要库
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m X� t"��Um��Computer Aided Geometric Design��G CAGD��"!QJ�7Evig6#�hE� tuRiÆArE�h4Qu%��G�Æ$=Jl�J�`�� CAGD u� �℄��&wP� �4Jl�J�`� u�U ,Ta�T0� ^ �0��Kt%|+h�1. JlJ�uv;� ^�U`��NC(Numerical Control) �6Tt:[��vkG:>G�5;�%�Q��=BiN�teu+`� O>3S�%*v;Jl�J�uP�+`�!MJl�J�u+`�5�#\�2\B;w℄>X;�Jl�J�Q ,v;Jl�J���P.Rd��uv;Jlu ,���1�v;Jlu Bezier , ��\ �f�X-��u;�Jl� I��b℄ Bezier Jl�[; Bezier Jlu��℄,tq ,Jlu?teT�te�^Æ� R1v; ,Jl��2�[;�3Jl>�uv;Jl ,���[;�3JlTMJl�M)V:`%3�u>XJl�'Jl6M>�MJluv;Jlhu9�{� Æ ,v;Jl�P �[T!!fqd� �u<O{�.�y����>G5;BGkeuv;J�u ,�2. ^6L5; �=B{�T0u �U�#��G?|-. uv;Jl�J�u�.�}('?|v;�. uv;�tTv;;W�U�G|IG.v;Jl�J���P[;Jlh1{u?teT�te%�Iu67.*aQP�v;�t�Æ2R%�G.';Jlu��.�.^\rTh,m5u,w��41x ��JsTu����>Gf J.steiner BBt�luAv;l%G�uu%0� �X�3. Bezier Jl�J�us#T#b� �^�U`� �W� u����s#T#b� ��,5;B Bezier keuJ��iAr%aguKE/�tqMJ�u�(�l� ,��P�J�uKE/+��(?Y;� �2Rf'�uX-#W 'o�u�.�y�qd� �uF`a��}KE/uW 'o��aBM Bezier J�um5u`�'o��Ron Goldman [;$�s�5-IrP���.f%��u>X Bezier Jl��P"y�fqU>X Bezier JluF`s#a��qB�4 ,JlJ�uF`��W>-.u�
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4. b�J��$=J�`�u<>^u6:�Catmull-Clark b�J���b℄ B�3J�>GqX-�-E/h�U�6M(jmi�{u)��[;K{u%#���tq Catmull-Clark J�uKE/uas�Æ��>oR%0� b�[KE/q Catmull-Clark J�u;W8��_q?�JlTJ�� ,�a�b�
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厦门大学博硕士论文摘要库
Abstract
Computer Aided Geometric Design (CAGD) is a subject which emerged with the
development of modern industry and computer science. Free-form curves and surfaces
modeling is one of the most important tasks in CAGD. This dissertation focuses on solv-
ing geometric approximation and convergence problems in curves and surfaces modeling.
The major contributions of this dissertation are summarized as follows.
1. Constant radius offsetting for curves and surfaces is one of the most important
geometric operations in CAD/CAM due to its immediate application to NC machining.
Due to the square root function in the denominator of unit normal vectors generally, the
exact offset curves and surfaces are not rational. Therefore approximations are needed,
often by using rational parameter curves and surfaces with low degree. This dissertation
presents two new methods of offset approximation: (1) Bezier Approximation algorithm
of offset curves. This algorithm firstly translates arbitrary parameter curves into the
piecewise cubic-degree Bezier curves, then using the properties of Bezier curve we can get
the tangent vector and the normal vector of every point on the approximation curve,
and calculate the approximating offset curve. (2) The approximating offset curve by
interpolatory using spline curve. Spline curve and base curve are combined to generate
a new rational curve by adding the weight. This curve approximates offset curve by
interpolating some sample nodes on the offset curve. We analyze and compare the
advantage and the weakness of these two algorithms, and apply the second method to
the approximation of tensor product offset surface.
2. In order to solve practical problems in real engineering applications, the classic
definition of offset curve should be extended, which means the fixed distance and direc-
tion in the classic definition will not be a necessary request. This dissertation presents
a definition of general offset curve, which has fixed offset distance, but variable offset
direction. The offset direction is defined by the local coordinate system, which is formed
of the tangent vector and the normal vector of every point on the curve. Based on this
definition, the curvature�regular and integral properties of the general offset curve are
III
厦门大学博硕士论文摘要库
discussed. As a result, J.steiner’s celebrated theorem of the oval is developed.
3. The algorithms of degree elevation and subdivision for Bezier curves�surfaces
play an important role in geometric modeling. For the Bezier tensor product surface,
recursive degree elevation and subdivision both generate a sequence of control meshes
that converge to the underlying Bezier surface, and get the piecewise bilinear approx-
imation of the original surface. This dissertation uniformly parameterizes the control
nets, provides the definition of its discrete partial derivatives for arbitrary order and
proves the smooth convergence property of these two algorithms. That is, the discrete
partial derivatives of control nets convergent to its corresponding continuous partial
derivatives. Ron Goldman introduced an alternative notion of rational Bezier curves
defined in terms of the negative degree Bernstein blending functions. This dissertation
proves the smooth convergence property of degree elevation for this kind of rational
Bezier curve as well. It is significative for the smooth property of approximating curves
and surfaces.
4. Subdivision surfaces are powerful and useful technique in modeling free-form
surfaces. The Catmull-Clark subdivision surface was designed to generalize the bi-cubic
B-spline surface to the meshes of arbitrary topology. By introducing the concept of
neighbor points and using the first-order difference of control points of Catmull-Clark
surfaces, we obtain the rate of convergence of control meshes of Catmull-Clark surface.
With the result of convergence we derive a computational formula of subdivision depth
for Catmull-Clark surface.
Key Words curves and surfaces; approximation; convergence; subdivision
IV
厦门大学博硕士论文摘要库
� z<lW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I^<lW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IIIHYn O�§1.1 �1(j. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
§1.2 v;JlTJ� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
§1.3 s#T#b� �uF`a� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
§1.4 Catmull-Clark b�J� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4HPn Fy�I���D3u§2.1 v;Jlu Bezier , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
§2.2 [;�3Jl>�uv;Jl , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
§2.3 [;�3J�>�uv;J� , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16H�n Fy�ID5be_a§3.1 �.T�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
§3.2 u��� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
§3.3 G. J.steiner �X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27H.n �I���D`h)�M§4.1 �1{| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
§4.2 s#T#b� �uF`a� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
§4.3 >X Bezier JluF`s#a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39H?n Catmull-Clark CT��§5.1 �.T�S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
§5.2 Catmull-Clark J�uKE/uas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
§5.3 b�[KE/q Catmull-Clark J�u;W8� . . . . . . . . . . . . . . . . . . . . . . . 547|<G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57�phYN5'Q:�o7<Dd℄Q+�< . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63uK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
厦门大学博硕士论文摘要库
Contents
Chinese Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .I
English Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
1 Introduction
§1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§1.2 Offset curves and surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§1.3 The smooth convergence of degree elevation and subdivision . . . . . . . . . . . . . . 3
§1.4 Catmull-Clark subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Approximation of offset curves and surfaces
§2.1 Bezier Approximation of offset curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
§2.2 The approximating offset curve by interpolatory using spline curve . . . . . . 11
§2.3 The approximating offset surface by interpolatory using spline surface . . . 16
3 The general offset curve and its application
§3.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
§3.2 Integral properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
§3.3 The theorem of general J.steiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 The smooth convergence for curves and surfaces
§4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
§4.2 The smooth convergence of degree elevation and subdivision . . . . . . . . . . . . . 32
§4.3 The smooth convergence of degree elevation for rational Bezier curves . . . 39
5 Catmull-Clark subdivision
§5.1 Definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46
§5.2 The rate of convergence of control meshes of Catmull-Clark sruface . . . . . . 47
§5.3 The computational formula of subdivision depth for Catmull-Clark surface 54
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Major Academic Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
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厦门大学博硕士论文摘要库
Degree papers are in the “Xiamen University Electronic Theses and Dissertations Database”. Fulltexts are available in the following ways: 1. If your library is a CALIS member libraries, please log on http://etd.calis.edu.cn/ and submitrequests online, or consult the interlibrary loan department in your library. 2. For users of non-CALIS member libraries, please mail to [email protected] for delivery details.
厦门大学博硕士论文摘要库