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Research on Approximation and Convergence

Problems in Curves and Surfaces Modeling

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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


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


� 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


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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�0 ��&��IQo Gz�J��E4v§2.1 Gz�JE Bezier 4vv;Jl�J�^�U`��NC(Numerical Control) �6Tt:[��vkG:>G�5;�1 +�v;Jl�*r�uT0�ÆB ,+�v;Jlu�4�)0>f �%L��P2R%�+�v;Jlu Bezier , ��q� , �!I}C�i�,B{i��^ ��:>!Ru��39\ ��fH*�X-��u;�Jl� I��b℄ Bezier Jl�[; Bezier Jlu��℄,tq ,Jlu?teT�te�Æ�te)EK{>B� ÆH*� 1v;Jl�Y. �IDTW=+�v;Jlu�.H{�HG�3+�Jlu offset Jl�#�fGtMJlhu�te�ÆBX-��u;�Jl� GtJlhu�te.4��%�℄,u��%*�U�H*fJl+��.�Q[�1� I��b℄ Bezier Jl�BezierJlu?teT�te℄,Gt�Æ��"!!�i��.{u�K �y^��JlhJs' y��Oi��H*39JlhJsu' Q+��.�=BJs� u#\��U�[;Jlh?teu('QP��.{�m;�JlI r(t)�K;�6CI ε1, [;��Go8�>

r′

(t) ≈r(t + ε1) − r(t)

ε1lK;�6CI ∆t�"Jlh%Ku9[d{ r(t)�r(t + ∆t) V?te�� αuCg� cos α�cos α =

r′

(t + ∆t) · r′

(t)

|r′(t + ∆t)| · |r′(t)|K8� ε2� L cos α < ε2�}�� α(0 < α < π2) dB�0�z�� { r(t + ∆t) -I%0�.{�7z �y;W |r(t + ∆t) − r(t)| I!yu��*\U>�H*P�Jlh%>u�.{�;�6C ∆t u�Kt!!y�H*�yJlhJs' dux��i��.RQ�

6


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= 1 Jl�u�0K�{uP�m��Jlud0�{I r(t1)�r(t2)�q�� ℄P�d0K{ P1�P2 }H�H��Jl�zs�tIJlh1{V?teux�t�U�^Jlh{ r(t1) Vux?�thK{ P1� { r(t2) u�?�thKq{ P2�.�~t|r(t1) − P1| = |r(t2) − P2| =

1

3|r(t2) − r(t1)|.m g = 1

3|r(t2) − r(t1)|� P1�P2 :1u� 8�^

P1 = r(t1) +r′

(t1)

|r′(t1)|· g, P2 = r(t2) −

r′

(t2)

|r′(t2)|· g. (3)39b℄ Bezier Jl8�

F (t) =3

∑

i=0

B3i (t)bi,1 B3

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7


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6 r 厦门大学博硕士论文摘要库3. the algorithms of degree elevation and subdivision for...

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