Shape Interrogation for Computer Aided Design and Manufacturing

422 Pages·2010·9.14 MB·English

by  Nicholas M. Patrikalakis

Shape Interrogation for Computer Aided Design
and Manufacturing-
Nicholas M. Patrikalakis Takashi Maekawa
Shape Interrogation for Computer
Aided Design and Manufacturing
Springer
Nicholas M. Patikalakis Takaslli Maekawa
Massachusetts Institute of Technology Department of Mechanical Engineering
Department of Mechanical Engineering Yokohama National University
Room 5-230A 79-5. Tokiwadii,
77 Massachusetts AN critic Hodogaya, Yokohama 240-8501
Cambridge, MA 02139-4307 Japan
USA maekawa@ 'ynu.acjp
nrnp@n it.edu
ISBN 978-3-540-42454-3 (hardcoUver) e-ISBN 978-3-642-04074-0
ISBN 978-3-642-04073-3 (softcover)
DOI 10.1007/978-3-642-04074-0
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: i..2009939440
I1 Springer-Verlag Berlin Heidelberg 2002, First softcover printing 2010
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t
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o
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Preface
Objectives and Features
Shape interrogation is the process of extraction of information from a geo-
metric model. Shape interrogation is a fundamental component of Computer
Aided Design and Manufacturing (CAD/CAM) systems and was first used in
such context by M. Sabin, one of the pioneers of CAD/CAM, in the late six-
ties. The term surface interrogation has been used by I. Braid and A. Geisow
in the same context. An alternate term nearly equivalent to shape interro-
gation is geometry processing first used by R. E. Barnhill, another pioneer of
this field. In this book we focus on shape interrogation of geometric models
bounded by free-form surfaces. Free-form surfaces, also called sculptured sur-
faces, are widely used in scientific and engineering applications. For example,
the hydrodynamic shape of propeller blades has an important role in marine
applications, and the aerodynamic shape of turbine blades determines the
performance of aircraft engines. Free-form surfaces arise also in the bodies of
ships, automobiles and aircraft, which have both functionality and attractive
shape requirements. Many electronic devices as well as consumer products
are designed with aesthetic shapes, which involve free-form surfaces.
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When engineers or stylists design geometric models bounded by free-form
surfaces, they need tools for shape interrogation to check whether the de-
signed object satisfies the functionality and aesthetic shape requirements.
0
+
This book provides the mathematical fundamentals as well as algorithms for
`
various shape interrogation methods including nonlinear polynomial solvers,
)
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intersection problems, differential geometry of i'ntersection curves, distance
functions, curve and surface interrogation, umbilics and lines of curvature,
geodesics, and offset curves and surfaces.
The book can serve as a textbook for teaching advanced topics of geome-
tric modeling for graduate students as well as professionals in industry. It has
been used as one of the textbooks for the graduate course "Computational
Geometry" at the Massachusetts Institute of Technology (MIT). Currently
there are several excellent books in the area of geometric modeling and in
the area of solid modeling. This book provides a bridge between these two
areas. Apart from the differential geometry topics covered, the entire book is
based on the unifying concept of recasting all shape interrogation problems
to the solution of a nonlinear system.
VI Preface
Structure and Outline
Chapter 1 presents a brief overview of analytical methods for the repre-
sentation of curves and surfaces in a computer environment. We focus on the
parametric representation of curves and surfaces, commonly used in CAD
systems for shape specification. We next review the theory of Bernstein poly-
nomials and associated algorithms and their application in the definition and
manipulation of Bezier curves and surface patches. Finally in this chapter,
we review the theory of B-spline basis functions and associated algorithms
and their application in the definition and manipulation of B-spline and Non-
Uniform Rational B-spline curves and surface patches. In our development of
Bernstein polynomials and B-spline basis functions and the associated curve
and surface representations, we do not provide detailed proofs as they are
already contained in other existing books on geometric modeling, on which
we rely for instructional purposes.
Chapters 2 and 3 provide an overview and introduction into the classical
elementary differential geometry of explicit, parametric and implicit curves
and surfaces, necessary for the development of the more advanced differen-
tial geometry topics that are presented in Chaps. 6, 8, 9 and 10. Much of the
material of Chaps. 2 and 3 (except the treatment of curvatures of implicit sur-
faces) can be generally found in various forms in existing books on differential
geometry and is included for convenience of the reader and completeness of
our development.
Chapter 4 focuses on the development of geometrically motivated solvers
for nonlinear equation systems and the related numerical robustness (relia-
bility) issues. Much of the shape interrogation problems defined and solved
in this book can be reduced to solving systems of n nonlinear polynomial
equations in 1 unknowns, each of which varies within a known interval. Much
p
of the development is based on the Interval Projected Polyhedron (IPP) al-
a
gorithm, developed in our Design Laboratory at MyIT in the early nineties.
Some shape interrogation problems involve more general nonlinear functions
including radicals of polynomials. These are also converted to nonlinear po-
lynomial systems of higher dimensionality via an auxiliary variable method.
The fundamental feature of the IPP algorithm is that it allows recasting of
continuous shape interrogation problems encountered in geometric modeling
and processing of free-form shapes into the discrete problem of computing
n'
convex hulls of a set eof points in a plane and their intersections with other
t
convex hulls along a particular axis. In this way, a bridge between the largely
disparate fields of geometric modeling of free-form shapes (largely based on
numerical analys-,4is and approximation theory) and discrete computational
geometry (largely based on the theory of algorithms and combinatorics) is
established. Another fundamental feature of the IPP algorithm, is the use of
rounded interval arithmetic motivated by questions of numerical robustness
or reliability, which have high importance in CAD/CAM systems. Interval
methods are a special branch of numerical analysis, with great potential for
Preface VII
applications in geometric modeling and processing problems. Interval me-
thods have not yet been used extensively in practice, because, if they are ap-
plied naively, they lead to interval growth that reduces the possible achievable
precision in a numerical computation. However, when combined with geome-
-
-
'
tric modeling algorithms based on convex combinations (as the de Casteljau
~'
r
algorithm), they lead to very minor interval growth and permit effective and
-'high accuracy solutions in practice. The IPP algorithm robustly eliminates
)
subregions of the domain which do not contain roots, thereby allowing effec-
tive bracketing of the roots of the nonlinear system within a given box with
size typically much smaller than the actual accuracy of the results of current
CAD/CAM systems.
Chapter 5 presents the first major shape interrogation problem analyzed
cad
in this book. Intersection is a fundamental operation in the creation of geome-
X
tric models encoded in the Bo7undary Representation form of solid modeling.
5
Intersection is also very useful in geometric processing for visualization, ana-
lysis and manufacturing of solid models. We present a unified methodology
4
0
for solving intersection problems, which reduces all such problems to solving
0 b
:.0
a system of nonlinear polynomial equations which in turn can be solved using
the method of Chap. 4. We also present a novel classification of intersection
problems by virtue of their dimensionality, the type of geometric representati-
c
ons involved, and the number system usred in problem statement and solution.
CAD a
The point to point, point to curve, point to surface, curve to curve, curve
to surface and surface to surface intersection problems are treated in some
+
c-
detail. Various special cases of interest, where the geometric entities involved
(points, curves and surfaces) are represented implicitly or parametrically in
terms of polynomials, are treated in some depth.
Chapter 6 is motivated by efficient tracing of intersection curves of two
surfaces which intersect either transversely or tangentially, and presents the
first, second and higher order derivatives of these entities for use in develo-
ping efficient and robust tracing algorithms. The surfaces involved may be
parametric or implicit in any combination.
Chapter 7 presents methodsh for the computation of the stationary points
-
,
of distance functions between points, parametric curves and parametric sur-
face patches (in any combination). The curves and surfaces may be defined
by general piecewise rational polynomials. The resulting problems are again
reduced to solving systems of nonlinear equations which can be solved using
the IPP algorithm developed in Chap. 4. Distance functions are closely rela-
ted to intersection problems and are also useful in many other applications
including feature recognition via medial axis transforms, animation, collision
detection, and manufactured object localization and inspection.
Chapter 8 addresses a variety of curve and surface interrogation methods
involving their position vectors and several higher order derivatives. Particu-
lar emphasis is placed on robust extraction of stationary points of curvature
maps and the consequent application in robust contouring of such maps.
VIII Preface
Again the problem reduces to solving systems of nonlinear equations which
4
Y
(
can be solved using the IPP algorithm developed in Chap. 4. The interroga-
tion methods analyzed in this chapter have many applications in aesthetic
and functional surface design and analysis, in fairing of oscillatory shapes, in
meshing of surface patches and in machining automation.
Chapter 9 discusses the problems of umbilics and lines of curvature as
methods of shape interrogation and identification. Umbilics are computed
via solution of a nonlinear polynomial system following the IPP algorithm of
Chap. 4. Curvature lines are computed via integration of a system of differen-
tial equations via an adaptive numerical process with specialized treatment
near umbilics. The stability problem of umbilics under perturbation of the
underlying surface is also analyzed for use in surface identification and feature
recognition probslems.
c
Chapter 10 addresses yet another shape interrogation problem involving
a
the geodesics of parametric and implicit surfaces. The classical geodesic equa-
tions are reviewed and numerical methods for the effective computation of
geodesics between two points on a surface or a point and a curve on a surface
are presented. The numerical methods involve iterative solution of a nonlinear
boundary value problem via either shooting or relaxation methods. Geode-
sics have applications in feature recognition via medial axis transforms, in
path planning in robotics (for distance minimization), in representation of
geodesic offsets for design and in manufacturing.
Chapter 11, the final chapter of this book, focuses on the problem of
b
offset (or parallel) curves and surfaces. Offsets have importantb applications
in NC machining, feature recognition via medial axis transforems and in to-
lerance region specification. The definition and computation of singularities
(and especially self-intersections) of planar offset curves and offset surfaces
is treated in depth. The methods developed are in part analytical, and in
part numerical relying on the IPP algorithm of Chap. 4 and on integration of
systems of nonlinear differential equations. The related concepts of Pythago-
rean hodographs, general offsets and pipe surfaces, which build on the theory
of offset curves and surfaces, are also reviewed and analyzed in some detail.
Problems that instructors can use in developing their own courses are
provided immediately after Chap. 11. Many of these problems have been
used in our graduate course at MIT.
Errors
A book of this size is likely to contain omissions and errors. If you have
any constructive suggestions or find errors, please communicate them to N.
M. Patrikalakis, MIT Room 5-428, 77 Massachusetts Avenue, Cambridge,
cad
MA 02139-4307, USA (e-mail: nmp(Qmit.edu), and T. Maekawa, MIT Room
5-426A, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA (e-mail:
[email protected]).
P0reface IX
1-
Acknowledgements
We wish to recognize the following former and current students who have
helped in the development of this book: Panos G. Alourdas, Christian Bliek,
Julie S. Chalfant, Wonjoon Cho, Donald G. Danmeier, H. Nebi Gursoy, An-
dreas Hofman, Chun-Yi Hu, Todd R. Jackson, Kwang Hee Ko, George A.
Kriezis, Hongye Liu, John G. Nace, P. V. Prakash, Guoling Shen, Evan C.
Sherbrooke, Stephen Smyth, Krishnan Sriram, Seamus T. Tuohy, Marsette
A. Vona, Guoxin Yu and Jingfang Zhou. We also wish to acknowledge Ste-
C
',
phen L. Abrams for his assistance with software development and Fred Baker
for editorial assistance.
We also thank Chryssostomos Chryssostomidis, David C. Gossard, Mal-
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colm Sabin, Takis Sakkalis, Nickolas S. Sapidis, Franz-Erich Wolter and Xiuzi
cad
Ye and several anonymous referees selected by Springer for useful discussions
and their comments.
We also wish to acknowledge MIT's funding of this book development
d o
from the Bernard M. Gordon Engineering Curriculumd Developmen+t Fund
'
via the Dean of the School of Engineering and via addoitional support from
the Department of Ocean Engineering.
We, finally, dedicate this book to our families, our wives Sandra Jean
and Yuko and our children Alexander, Andrew, Nikki, and Takuya, whose
love, patience, understanding and encouragement made this lengthy project
possible.
Cambridge, MA, June, 2001 Nicholas M. Patrikalakis
Takashi Maekawa
Contents
...................
Representation of Curves and Surfaces
1.
......................... 1
1.1 Analytic representation of curves 1
1.1.1 Plane curves ..................................... 1
1.1.2 Space curves ............................................................. 3
1.2 Analytic representation of surfaces 4
...............................
1.3 Bezier curves and surfaces 6
............................
1.3.1 Bernstein polynomials 6
1.3.2 Arithmetic operations of polynomials in Bernstein form 7
1.3.3 Numerj'ical condition of polynomials in Bernstein form. 9
( .........
1.3.4 Definition of Bezier curve and its properties 12
.......................
1.3.5 Algorithms for Bezier curves 13
...................................
1.3.6 Bezier surfaces 018
.............................
1.4 B-spline curves and suryfaces 20
't7 ......ra..................................
1.4.1 B-splines 020
...................................
1.4.2 B-spline curve 21
.....................
1.4.3 Algorithms for B-spline curves O24
..................................
1.4.4 B-spline surface 29
.....................
1.5 Generalization of B-spline to NURBS 30
..........................
2. Differential Geometry of Curves 35
............................
2.1 Arc length and tangent vector 35
i-S ..........................
2.2 Principal normal and curvature 39
.............................
2.3 Binormal vector and torsion 43
..................................
2.4 Frenet-Serret formulae 47
.........................
3. Differential Geometry of Surfaces 49
.......................
3.1 Tangent plane and surfhace normal 49
ra ........................
3.2 First fundamental form I (metric) 52
..................
3.3 Second fundamental form II (curvature) 55
....................................
3.4 Principal curvatures 59
...........................
3.5 Gaussian and mean curvatures 64
..................................
3.5CSI.1 Explicit surfaces.................................. 64
3.5.2 Implicit surfaces 65
....................
3.6 Euler's theorem and Dupin's indicatrix 68