An Insight into Geometric Modelling and Curves and Surfaces

325 Pages·41.997 MB·English

by  Laksa A.

An insight into
geometric modelling
and
Curves and Surfaces
By
Arne Laksa˚
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iv
Preface
Geometry – from Ancient Greek, earth measurement – has been an important ingredient
ofthedevelopmentofscienceandlateralsoindustry,designandproduction.
In modern time, the Geometric modeling community is established, and has now a 50-
year-old history. The first pioneers were motivated by the introduction of computers in
design, construction and manufacturing. The goal was basically to provide methods and
algorithmsforcurveandsurfacerepresentationsandcalculationsonthese,andtocombine
curve and surface methods in computer graphics and simulations. By doing this, they
started the develop of a new discipline called geometric modeling, including computer
aidedgeometricdesign,solidmodeling,algebraicgeometry,andcomputationalgeometry.
Computer aided geometric design (CAGD) is the central part of the field. It started with
the development of methods and algorithms for CAD/CAM. Today, however, is support
for virtual reality/design and computer games, simulators and animations in movies and
TVproductionsasimportantareas.
Curvesandsurfacesforcomputeraidedgeometricdesignstartedwithclassicalgeometric
objects as line, arc, plane, sphere, cylinder etc. The next step in the development of free-
formgeometrywasBezier,Hermite,B-splineandrationalvariantsasrationalBezierand
NURBS and subdivision surfaces. The development began in earnest in the 1960s and
wentwithfullforceuntil1990. Itis,however,stillanongoingdevelopmentthatincludes
T-splines and other improvement as Multivariate splines. New results also includes the
introduction of generalized B-splines and thus it is still a long way to go. This way of
constructing curves and surface is the most important today and will probably be it as far
as we can see. It is the defacto industrial standard. What we can call a 2nd generation
curve and surface construction started around year 2000, and it is just in its initial phase.
Itincludesdifferenttypesofcurvesandsurfacesconstructedbyblendingtechnics.
For me the work with blending technics accelerated one day the summer of 2003. I went
to my college Børre Bang at his office, to discuss improvements of GM lib , a C++ open
source programming library for geometric modeling (see [88]). Lubomir Dechevsky was
already there. He asked if we could look at a function he believed could be the limit
of a polynomial B-spline when the degree (and, thus number of knots) tends to infinite
(describedin[44]). Sincewethenwereworkingwiththegraphicalpartofaprogramming
library GM lib , we implemented this new basis function and tried it for curves in R3.
The result was a piecewise linear curve just like 1st degree B-spline curves, but there
was, however, a big difference; this new curve was C∞-smooth but obviously only G0,
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i.e. piecewise linear. After a short discussion we replaced the coefficients of the curve
with something we called “local curves”, and Expo-Rational B-spline (ERBS) curves
were born. Later this has been expanded to a more general concept we call Generalized
Expo-RationalB-spline.
This book was first planned to be about blending technics. However, because of input
frombothstudents,thismanuscriptisalsoinfluencedbythefactthatpartsofithavebeen
used as compendiums in courses in a master program in Computer Science at Narvik
University College (NUC), and industrial cooperation partners I decided to expand the
book to include free-form geometry in general. Free-form geometry has attracted my
attention for the more than 20 years both as a scientist, teacher and also for industrial
implementations([89],[60]).
Acknowledgements
FirstIwillexpressmygratitudetomycolleaguesLubomirT.DechevskyandBørreBang
foraveryfruitfulcooperation. NextIamespeciallygratefultoTomLyche,KnutMørken
forveryvaluablediscussionsandusefuladvice.
I am also grateful to Tor Dokken, Bernt Bremdal, Joakim Gundersen, Arnt Kristoffersen,
JosteinBratlieandRuneDalmofordiscussions,helpingwithGM lib,documentationand
otherpracticalissues,andtoSusanJaneBerntsenforlanguageconsultations.
Finally,IwouldliketogivebigthankstomywifeMarit,forbearingwithmethroughthis
work.
December2012 ArneLaksa˚
Contents
Preface v
1 Introduction 1
1.1 Historicalnotesaboutgeometry . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Moderngeometry . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 Elementarygeometry . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 Non-Euclideangeometry . . . . . . . . . . . . . . . . . . . . . . 10
1.1.4 Introductionofmathematicalrigor . . . . . . . . . . . . . . . . . 11
1.1.5 Algebraicgeometry . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.6 newnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Geometricmodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 GM lib,anOpenSourceprogramminglibrary . . . . . . . . . . . . . . . 18
1.3.1 BLAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Algorithmiclanguage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6 Overviewofthisbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 MathematicalspacesusefulinCAGD 29
2.1 Euclideanspaces,cartesiancoordinatesandvectorspaces . . . . . . . . . 30
2.2 Homeomorphism,diffeomorphismandmanifolds . . . . . . . . . . . . . 31
2.2.1 Local/globalparametrization,chartsandatlas . . . . . . . . . . . 31
2.3 Compactspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Affinespace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 ProjectivespaceandGrassmannien . . . . . . . . . . . . . . . . . . . . 36
2.6 Homogenouscoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 38
I Curves 39
3 ParametricCurves 41
3.1 Differentiations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.1 Regularcurves-arclengthparameterization . . . . . . . . . . . 45
3.1.2 Reparameterization . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.3 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Basisfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
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viii CONTENTS
3.3 HermiteCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Be´zierCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.1 Bernsteinpolynomial . . . . . . . . . . . . . . . . . . . . . . . 57
4 SplineCurves 63
4.1 Historyofinterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 DivideddifferencesandNewtonpolynomial . . . . . . . . . . . 69
4.1.2 Lagrangepolynomial . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.3 Hermiteinterpolation . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Hermitespline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Cubicsplineinterpolation . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 HistoryofB-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 ModernB-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5.1 Be´ziercurvesanddeCasteljaualgorithm . . . . . . . . . . . . . 85
4.5.2 TheB-splinefactormatrixT(t) . . . . . . . . . . . . . . . . . . 87
4.5.3 CommutativityrelationsbetweenT(t)matricesandtheirderivatives 90
4.5.4 B-splinesonMatrixnotations . . . . . . . . . . . . . . . . . . . 91
4.5.5 Examples of B-splines, matrix notation, de Casteliau’s algorithm
andderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5.6 HermitesplineinterpolationonB-splineform . . . . . . . . . . . 96
4.5.7 CubicsplineinterpolationonB-splineform . . . . . . . . . . . . 97
4.5.8 B-splinesandknotinsertion . . . . . . . . . . . . . . . . . . . . 99
4.5.9 NURBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5.10 ImportantpropertiesofsplinefunctionsonB-splineform . . . . . 103
4.6 Blossoming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.7 Useofblendinginconstructionofgeometry . . . . . . . . . . . . . . . . 104
5 Blendingfunctions−definitionsandbasicproperties 107
5.1 Fourexamplesofdifferentblendingfunction . . . . . . . . . . . . . . . 107
6 “ERBS”−definitionsandbasicproperties 115
6.1 Asimpleversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2 Generalizeddefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3 Basicproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 TheScalablesubset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5 Thedefaultsetofintrinsicparameters . . . . . . . . . . . . . . . . . . . 134
6.6 Expo-RationalB-splinefunctions . . . . . . . . . . . . . . . . . . . . . . 139
6.7 Knotvectorsandcontinuity . . . . . . . . . . . . . . . . . . . . . . . . 142
6.8 HermiteInterpolationproperties . . . . . . . . . . . . . . . . . . . . . . 143
6.9 Influenceoftheintrinsicparameters . . . . . . . . . . . . . . . . . . . . 145
7 GERBS–Curves 151
7.1 Definition/implementationof“open/closed”curves . . . . . . . . . . . . 153
7.2 Evaluation,valueandderivatives . . . . . . . . . . . . . . . . . . . . . . 155
7.3 Be´ziercurvesaslocalcurves . . . . . . . . . . . . . . . . . . . . . . . . 158
7.3.1 LocalBe´ziercurvesandHermiteinterpolation . . . . . . . . . . 161
CONTENTS ix
7.3.2 Samplingandpreevaluation . . . . . . . . . . . . . . . . . . . . 162
7.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.4 Circulararcsaslocalcurves . . . . . . . . . . . . . . . . . . . . . . . . 171
7.4.1 LocalArccurvesandmodifiedHermiteinterpolation . . . . . . 171
7.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.4.3 Reparametrizationandusinganapproximativecurvelengthparametriza-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.5 Affinetransformationonlocalcurves . . . . . . . . . . . . . . . . . . . 182
II Surfaces 187
8 ParametricSurfaces 189
8.1 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.1.1 ThedifferentialdS . . . . . . . . . . . . . . . . . . . . . . . . . 192
p
8.1.2 Curvesonsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.1.3 ThetangentplaneT (S) . . . . . . . . . . . . . . . . . . . . . . 194
q
8.1.4 Firstfundamentalform . . . . . . . . . . . . . . . . . . . . . . . 195
8.1.5 Secondfundamentalform . . . . . . . . . . . . . . . . . . . . . 197
8.2 Surfaceofrevolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.3 Surfacebysweeping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.4 SurfacesbasedonCurveconstructions . . . . . . . . . . . . . . . . . . 203
8.5 Booleansumsurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.5.1 CoonsPatch,bilinearblending . . . . . . . . . . . . . . . . . . 205
8.5.2 CoonsPatch,bicubicallyblending . . . . . . . . . . . . . . . . . 206
8.5.3 Hermiteblendingsurface . . . . . . . . . . . . . . . . . . . . . . 209
8.5.4 Curvesontriangularsurfaces . . . . . . . . . . . . . . . . . . . . 220
9 TensorProductSurfaces 225
9.1 Definition/implementationof“open/closed”Surfaces . . . . . . . . . . . 227
9.2 Evaluation,valueandderivatives . . . . . . . . . . . . . . . . . . . . . . 229
9.3 Be´zierpatchesaslocalpatches . . . . . . . . . . . . . . . . . . . . . . . 235
9.3.1 LocalBe´zierpatchesandHermiteinterpolation . . . . . . . . . . 236
9.3.2 ExamplesofHermiteinterpolations . . . . . . . . . . . . . . . . 238
9.4 FreeformsculpturingusingtensorproductERBSsurfaces . . . . . . . . 245
9.5 Computational and implementational aspects for tensor product ERBS
surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10 TriangularbasedSurfaces 259
10.1 Homogenousbarycentriccoordinatesforsimplices . . . . . . . . . . . . 261
10.2 Be´ziertriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
10.3 ThegeneralsetofExpo-RationalB-splinebasis-functioninhomogeneous
barycentriccoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
10.4 ERBStriangles,definitionandevaluators . . . . . . . . . . . . . . . . . 269
10.5 LocalBe´ziertrianglesandHermiteinterpolation . . . . . . . . . . . . . 273
10.6 Sub-trianglesfromgeneralparametrizedsurfacesaslocaltriangles . . . . 284
x CONTENTS
10.7 Surfaceapproximationbytriangulations. . . . . . . . . . . . . . . . . . . 286
10.8 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Bibliography 299
ListofAcronyms 309
Index 311