toward a quantitative method for evaluating aesthetic...
TRANSCRIPT
, TOWARD A QUANTITATIVE METHOD FOR EVALUATING AESTHETIC QUALITIES IN
SHAPE USING DERIVATIVES OF GAUSSIAN AND NORMAL CURVATURES
1 + I 'Ngtional Library of Canada
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permission. autorisation.
APPROVAL
Name: Quan Liu
Degree: Master of Science
Title of Thesis: TOWARDA Q U A N T I T A T I ~ METHOD FOR EVAL-
UATING A E S T H E T I C Q U A L I T I E S IN S H A P E W S I N G
DERlVATIVES O F GAUSSIAN A N D N O R M A L CITR-
VATlJRES
Examining Committee: Dr. M. Stella Atkins
Chair
d J o h n C. Dill
Senior Supervisor
/
Dr. F. David Fracchia
Supervisor
Dr. John D. Jones
Examiner
Date Approved:
Abstract
For many real world objects such as automobiles, one major part of their attractiveness
of appearance lies in the smooth blending of highlights and shadows. This, in turn.
is determined by the geometric properties of surfaces. It is believed that there is a
relationship between mathematical smoothness and -aesthetical fairness of surfaces.
Most analysis methods for parametric surfaces focus on visualizing the geometric
properties by showing curvatures or characteristic curves. None of them has ever
succeeded in giving a quantitative measure for the fairness. In this thesis, obe method
is proposed to compute the quantitative fairness of parametric surfaces. C
We use normal curvature, Gaussian curvature and th& first and second derivatives
to explore the relationship between mathematical smoothness and aesthetical fairness
of parametric surfaces. We have implemented a tool to compute the first and second
derivatives of normal and Gaussian curvatures,'which give the quantitative fairness
of the parametric curve network. Surfaces can be sorted based on their quantitative
fairness. User testing is described to show that our method is effective in measuring
aesthetic qualities of parametric surfaces.
This method will help designers to decide the quality of surfaces and find design a
problems. 7
Acknowledgments
I would like t o express my greatest appreciation to my senior supervisor Dr. John
Dill, supervisor Dr. avid Fracchia for their guidance encouragement and academic
help throughout my thesis work. I am also grateful to Dr. John Jones for his careful
reading of this thesis and his valuable comments. My discussion'with my friends
Changbao Wu, Hongsheng Chin and Yongping Luo was irery helpful in my thesis
work. Finally. I acknowledge the financial support from the School of Computing
Science, Simon Fraser University.
Contents
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A b s t r a c t . . 111
.4cknowledgments . . . . . . . . . . . . . . . . . . . . . . . .* . . . . . . . . iv '8 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables v111
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 An Introduction to Surface Analysis . . . . . . . . . . . . . . . . . . . I
1.1 hlotivation of Surface Analysis . . . . . . . . . . . . . . . . . . 1
1.2 Related Work on Surface Analysis .) - . . . . . . . . . . . . . . . . 1.2.1 .I Planar Curve Fairness .)
1 " - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 2 2 Surface Fairness : 1
1 .2.3 Surface Analysis Methods . . . . . . . . . . . . . . . 1
. . . . . . . . . . . . . . 1.2.4 Surface Generation Methods 1 1 - . . . . . . . . . . . . . . . . . . 1.:3 Our Surface Analysis Method 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Differential Geometry 15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Curves 1.5
. . . . . . . . . . . . . . . . . . . . . . 2.1.1 Space Cuh.es 1.5
. . . . . . . . . . . . . . . . . . . . 2.1 2 Surface Curves .. 19
2.2 Surfaces . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . a>*) -- 2.2.1 The Equation of a Surface . . . . . . . . . . . . . . . 'P) - - 2 . 2 Surface Normal ')') - - . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 2 . 3 T h e First Fundamental Form 2 3
. . . . . . . . . . . . '> 4 The Second Fundamental Form 2:j -.-. 3
* . . . . . . . . . . . . . . . . 2.2..5 Surface Normal C'urvaiure 2-1 . -
' -.-. ' 6 Gaussian Curvature . . . . . . . . . . . . . . . . . . 2.3
i
. . . . . . . . . . . . . of Normal and Gaussian Curvatures 27
b m : e Fairness . . . . . . . . . . . . . . . . . . . . . . 27
. . . . . . . . . . . . . . . . . . . . . . . . . 3.2 ) Surface Fairness .. 29
. . . . . . . . . . . . . . . . . . . . . . Implementat. ion Issue ' a 30 a
3.3.1 Computing the First and Second Derivatives of Nor-
. . . . . . . . . . . . . . . . . . . . . I ma1 Curvature 30 .3 .3 .2 Computing the First and Second Derivatives of Gaus- . q
. . . . . . . . . . . . . . . . . . . . . sian Curvature 34
3.3.3 ComputingOAny Order Partial Derivative of Paramet-
. . . . . . . . . . . . . . . . . . . . . . . . ric Surfaces 39
. . . . . . . 3.3.4 Computing the Tangent Vector (du . dz. ) 39
3.33.5 Computing the Fairness of a Network of Isoparametric . . . . . . . . . . . . . . . . . . . . . . . . . . Curves 1 -10
. 3.3 .6 , Computing Planar Curves and Lines of Curvature . . 4 1
. . . 3.3.7 Computing the Maximum Change of I d j ~ , l ldsl 4 3
. . . . . . . . . 3.3.8 Computing the Integral of Idlrc, l ldsl 43
3.3.9 The Interface of Our Program SURF . . . . . . . . . 43
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Analysis 47
. . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Torus : . . . . . 1'7
. . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Planar Curves -18
. . . . . . . . . . . . . . . . . . 4 . 1 2 Isoparametric Curves -18
. . . . . . . . . . . . . . . . . . . -1.1.3 Lines of Curvature 49
. . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Summary 50
. . . . . . . . . . . . . . . . . . . . . . . . 4.2 Lid of Utah Teapot 50
. . . . . . . . . . . . . . . . . . . . . . 4.2.1 Planar Curves 50
. . . . . . . . . . . . . . . . . . . 4.2.2 Isoparametric Curves 50
. . . . . . . . . . . . . . . . . . . 4.2.3 Lines of Curvature 51
. . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Summary 51
1.4 Bump . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Improved Bump .5 3
. . i : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 More Tests 54
. . . . . . . . . . . . . . . . . . . . . . 4.6.1 Four Saddles 5.5
. . . . . . . . . . . . . . . 4.6.2 Three Telephone Handsets 5.5. 4
. . . . . . . . . . . . . . . . . . . 4.6.3 - Three Mouse Tops .55
. . . . . . . . . . . . . . . . . . . . . . . . 4.6.4 Four'Vases ri6
. . . . . . . . . . . . . . . . . 4.6.5 Four Perfume Bottles 56
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Discussion 8-1 . . . . . . . . . . . . . . 5.1 The Sample Correlation Coefficient r 83
. . . . . . . . . . . . . 5.2 The ~ o ~ u l a t i G n Correlation Coefficient p 85
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3 More Analys.is 87
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion , 92
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Summary 92
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Conclusion 93
. . . . 6 . 3 Fu tu rework f . . . . . . . . . . . . . . . . . . . . . . . . 93
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols 99
vii
List af Tables
Average number of segments of constant sign of the torus. . . . . . 49
Analysis result of the lid of Utah teapot. . . . . . . . . . . . . . . . 51
Average number of segments of constant sign for bump. . . . . . . . 57
Average number of segments of constant sign for improved bump. . .57
Testing result of two bumps. . . . . . . . . . . . . . . . . . . . . . . 57
Weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 - -
Fairness of bumps. . . . . . . . . . . . . . . . . '. . . . . . . . . . . :I r
Testing result of saddles. . . . . . . . . ., . . . . . . . . . . . . . . . .'>ti
-1.9 Average number-of segments of constant sign for saddles. . . . . . . 59
1.10 Fairness and average rank for saddles. . . . . . . . . . . . . . . . . . .59
4.11 Testing result of telephone handsets. . . . . . . . . . . . . . . . . . 60 2
4.12 Average number of segments of constant sign for telephone handsets. 60
4.13 Fairness and average rank for teleph'bne handsets. . . . . . . . . . . 60
4.14 Testing result of mouse tops. . . . . . . . . . . . . . . . . . . . . . . 61 b
4.15 Average number of segments of constant sign for mouse tops. . . . . 61
3.16 Fairness and mean rank for mouse tops. . . . . . . . . . . . . . . . . 61
4.17 Testing result of vases. . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.18 Averagenumberofsegme~~tsofconstants ignforvases . . . . . . . . 6:)
4.19 Fairness and average rank for vases. . . . . . . . . . . . . . . . . . . 6:3
4.20 Testing result of perfume bottles. . . . . . . . . . . . . . . . . . . . 6 4
4.21 AveragenumberofsegmentsofconstantsignforPerfumebottles. . 65
4.22 Fairness and average rank for perfume bottles. . . . . . . . . . . . . 6 5
5.1 Different average sampje correlation coefficients with and without per-
fume bottle # 4. . . . . . . . . . . . . . . . . '. . . . . . . . . . . . 89
List ,of Figures
1.1 Reflection lines: g f on ;c is t he mirror image of the straight line g. . . 6 -
1.2 Reflection line analysis 1. (Courtesy of Hagen) . . . . . . . . . . . . . I - 1.3 Reflection line analysis 2.' (Courtesy of Hagen) . . . . . . . . . . . . . I
1.4 A surface ( in t h e middle) with two focal surfaces. (Courtesy of Hagen) 9
1.5 A car hood with generalized focal surfaces. (Courtesy of Hagen) . . . 10
1.6 Trimmed surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 1'1
1.7 Co-continuous surfaces. (Courtesy d f Hagen) . . . . . . . . . . . . . . 14
1.8 C1-continuous surfaces. (Courtesy of Hagen ) . . . . . . . . . . . . '. . 1-1 - L
1.9 C2-continuous surfaces. (Courtesy of H a g e n ) . . . . . . . . . . . . . . 1-4
1.10 Color m a p of isophotes on C'3-continuous surfaces. (Courtesy of Hagr n ) 14 .
2.1 Position vectors- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5
2.2 T h e chord vector PQ, . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Space curve frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 *
2.4 A curvature profile ~ ( s ) i f ~ ( s ) is always positive. . . . . . . . . . . . 18
2.5 Redefined curvature profile, ~ ( s ) signed. . . . . . . . . . . . . . . . . 1S
2.6 The surface curve frame. . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 T h e angle 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1
2.S Elliptical surface. tic > 0. . . . . . . . . . . . . . . . . . . . . . . . . 20
2.9 Hyperbolical surface: tic < 0. . . . . . . . . . . . . . . . . . . . . . . 26
2.10 Ql ind r i ca l surface, K G = 0. . . . . . . . . . . . . . . . . . . . . . . . 26
3.1 Two curves ( 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Two curves (2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 s
* .I 3.3 Two curves (3 ) . . . :
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Surface mesh : . 41 L
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Parametric space 41
3.6 T h e interface of SU,RF ( 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . " 4.5
. . . . . . . . . . . . . . . . . . . . . . . . 3.7 T h e intdrface of S U R F ( 2 ) : 46
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Torus. tic 66
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Torus. P l l o g / t i c I I l /ds2 66
. . . . . . 4.4 Torus. ti,. . Planqr curves are perpendicular t o r.axis : . . 66
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' $-5 Torus. d l t i n x I / d s . 67
. . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Torus.621logItin1Il/ds2. 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Torus. sign of tic
1.8 Torus. sign of d l t i c , I / d s . . . . . . . . . . . . . . . . . . . . . . . . . . 67 . . . . . . . . . . . . . . . . . . . . . . . 4.9 Torus. sign -of 62 ( logl t iGI I ( I d s 2 67
. . . . . 4.10 Torus.. sign of ti,, . Planar curves are perpendicular t o s.axis 68
. . . . . . . . . . . . . . . . . . . . . . . . . 4:11 Torus. 'sign of dlti,, ( I d s 68 -4
. . . . . . . . . . . . . . . . . . . . . . 4.12 Torus . sign of 8 Iloglti,, I l /ds2 68
. . . . . . . . . . . . . . . 4.13 Torus. parametric d i r e t i o n of constant u 68 s
. . . . . . . . . . . . . . . . . 4.13 Torus. parametric dirPttion of constant 1 ) 68
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Torus . ti,, 69
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Torus. K , ~ 6'3
4.17 Torus. d ( t i nu lids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Torus. dl t in t . l /ds . 69
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Torus . 62110gl~, , l l /ds2 69
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 Torus . 8110gltin. l l / d s 2 69
4.21 Torus. sign of K,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.22 Torus. sign of tint 70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Torus. d l ~ ~ , \ I d s 70
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.24 Torus. 62 IlogItiGU I I / d s2 70 'a
. . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Torus. sign of d l ~ ~ , l / d s 70
. . . . . . . . . . . . . . . . . . . . . . . . 4_26 Torus. sign of @ l l o g l ~ c u 1 l / ds2 70
. . . . . . 4:27 Lid . sign of K,. . Planar curves are perpendicular to x.axis 71 I
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.28 Lid. sign of dl knX [ I d s i l
. . . . . . . . . . . . . . . . . . . . . . 4.29 Lid. sign of 8 IloglrtnX I J / d s 2 . 71
4.30 Lid. parametric direction of constant u . . . . . . . . . . . . 4.31 Lid. parametric direction of constant t7 . . . . . ; 71
4.32 Lid. sign of K,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.33 Lid. .sign of tinv 72
4.34 Lid. sign of d ( ~ , , I /ds . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.35 Lid. sign of d l ~ , . I /ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.36 Lid. sign of d2 l l og (~ , , I l / ds2 . . . . . . . . . . . . . . . . . . . . . . . * 72
4.37 Lid. sign of 6211ogl~,, I l /ds2 . . . . . . . . . . . . . . . . . . . . . . . 72 . . . . . . . . . . . . . . . . . . . . 4.38 Lid. maximum principal direction 713
. . . . . . . . . . . . . . . . . . . . . 4.39 Lid. minimum principal direction 73
4.40 Lid . sign of K,,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.41 Lid, sign of K,., . 7.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.42 Lid, sign of dl~,,,l/da 7:3
. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.43 Lid, sign of dl~, , , l /ds 713
. . . . . . . . . . . . . . . . . . . . . . . 1.44 Lid, sign of d2110glti,,,,l/ds2 7-1
. . . . . . . . . . . . . . . . . . . . . . . . 4.45 Lid. sign of d2110gl~,,, l l / ds2 7-1
4.46 Pat.2, parametric direction of constant u . .-, .
4.47 Pat2. sign of K , ~ . . . . . . . . . . . . . . . . . . . . . . . * . . . . . . 1.1
.-, - . . . . . . . . . . . . . . . . . . . . . . . . . . 4.48 Pat2, sign of d l ~ , [ I d s I : )
. . . . . . . . . . . . . . . . . . . . . . . 4.49 Pat2. sign of . 1.2
. . . . . . . . . . . . . . . 4 ..i 0 Bump. parametric direction /of constant u 76
4 ..5 1 Bump . K,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.52 Bump. d l ~ , , I /ds 76
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.53 Bump. 61 I l o g l ~ , I l / ds2 76
4.S4 Bump, sign of K n u . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 .-, - . . . . . . . . . . . . . . . . . 4 ..5 5 Bump . sign of d l ~ ~ ~ 1Id.s. . + . . . . . . . 1 1
- - 1 ..5 6 Bump sign of d 2 ) l ~ g l t i n u l l / d s2 1 1
< . . . . . . . . . . . . . . . . . . . . . . .
xii
-- . . . . . . . . . 4..57 Improved bump, parametric direction oT constant u . I r - - . . . . . . . . . . . . . . . . . . . . . . . . . . 4.58 Improved bump, nnU. r I . . -- . . . . . . . . . . . . . . . . . . . . . . . 4..59 Improved bump, dlti,, I / & . I r
3.60 Improved bump, 61IloglnnU ( l / d s 2 . . . . . . . . . . . . . . . . . . . . 7s . . . . . . . . . . . . . . . . . . . . . . 4.61 Improved bump. sign of rnu. iS
. . . . . . . . . . . . . . . . . . . 4.62 Improved bump, sign of d ( ~ , , ( I d s . i S
4.63 Improved bump, sign of 61 l l o g l ~ , , 1 1/ds2. . . . . . . . . . . . . . . . 7 8 -
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "4.64 Bump. 75
. . . . 4.65 Improved bump. ., . . . . . . . . . . . . . . . . . . . . . . . . 78
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.66 Saddle 1. 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.67 Saddle 2. 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.68 Saddle 3. 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.69 Saddle 4. 7!J
. . . . . . . . . . . . . . . . . . . . . . . . . 4.70 Telephone+handset 1. .: $0
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.71 Telephone handset 2. $0
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.72 Telephone handset 3. SO
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.73 Mouse top 1. 81
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.74 Mouse top 2.. 81
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.75 hlouse top 3. S I
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.76 blase 1. $2
4 .77Vase2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SL)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.78 Vase 3. #2
4.79Vase4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.80 Perfume bottle 1. . . . , . . . . . . . . . . . . . . . . . . . . . . . . . 83
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.81 Perfume - bottle 2. $1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.82 Perfume bottle 3. $:3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.&3 Perfume bottle 4 . 811
. . . . . . . . . . . . . . . . . . 1 Fairness and average rank for saddles. 89
. . . . . . . . . . . 5.2 Fairness and ai.erage rank for telephone handsets. 90
. . . . . . . . . . . . . . . 5 . 3 Fairness and average rank for mouse tops. 90
... X l l l
Chapter 1
An Introduction to Surface
Analysis
a 1.1 - -. Motivation of Surface A-nalysis
In technical design. the curves and surfaces that one designer encounters can usuallj.
bc classified into two categories: aesthetic (such as &utomobile bodies) and functional
(such as airplane fuselages) [ lo] . In this thesis, I will concentrate on the first categorj..
although often both requirements tend to go hand in hand.
For many real world objects such as automobiles, one main part of their attractive-
ness lies in the smooth blending of highlights and shadows. This, in turn, is determined
the geometric properties of surfaces. Thus a major part of the computer-aided dc-
sign is to construct mathematically smooth. aesthetically pleasing surfaces [6].
Design problems can be found when the objects are finally produced. or when
the curves or surfaces .are plotted t o full scale on a large flatbed plotter. Rut hot h
methods are t ime consuming and expensive. thus it is impertant tha t the designer
find design problems first using the CAD terminal. Because of the physical l imitatio~is
of the screen. even i f two curves or surfaces look identical (using shading methods).
t h e may have significant shape differences ~vhen they are at full scale. HOW to rnakv
potential design errors visible to the designer on the terminal is one important goal
of surface analysis in C.4D.
CHAPTER 1 . AN INTRODljCTION TO SURFACE ANALYSIS s> -
To achieve this goal, the fundamental work of surface analysis is t o define the
criteria of surface fairness in a mathematical way so that we can make full use of the
power of computers to measure the aesthetic quality of parametric surfaces and det,ect
potential design problems.
1.2 Related Work on Surface Analysis
~ e f o r e we discuss surface fairness, it is important to understand the criteria of planar
curve fairness. One major reason is that surfaces can be regarded as being composed
of numerous planar curves.
\
1.2.1 Planar Curve Fairness
The earliest literature discussing planar curve fairness mathematically is by Birkhoff
(31 :
e A first obvious requircment, therefore, is that the curz!ature varies gradually (that
is , continuously) along the curve and oscillates as few times as possible zn ~ t i c u *
of the prescribed characteristic points and tangents.
A second like requircment is that the m a ~ i m u m rate of changc of curz~aturc bf
as small as possible along the contour.
Birkhoff developed these criteria based upon his study of vases.
In 167. D ~ I I suggests:
1,
.-I curve is fair if it consists of a small number of regions where thc cumatur.t
changes monotonically and smoothly.
Dill developed this definition during his work on computer graphics for automobile
design at General l iotors Corporation. I(
In [XI]. Su and Liu suggest that a planar curve be called fair if the following three
conditions are satisfied:
C H A P T E R 1. A N ' INTRODUCTION T O SURFACE ANALYSIS
1
, The curve is ojGC2-continui ty;
there are no unwanted inflection-points on the curve;
9a s 0 the curvature of the curve yaries in an even manner.
/'r More specifically. t he third q n d i t i o n has the following implications:
w
The number of rrtreme points of the curvature should be as small as possible.
the curziature of the curzTt between two adjacent extreme points should vary al- \
most linearly.
Su and Liu base these criteria on their experience in developing CAD/CAM systems
at ship building factories in Shanghai, China. ,
1.2.2 Surface Fairness
Unlike planar curve fairness, there are relatively few papers which discuss conditions
for surface fairness. In [35], Su and Liu give one suggestion for defining surface fairness:
0 If every curve, which is the cross-stction of an arbitrary plane and a surfact. I,\
fair, then the surfact 1s said to be fair.
r
.4s'pointed out in [35]. it is impossible to check whether or not every curve is fair. In
practical applications, to justify whether a surface is fair or not, only a few- families
of network curves on the surface are examined. One quest#5n about this definition is
whether we should treat planar curv'es as space curves or surface curves (for definitions P
of space curve and supface curve, please refer to Section 2.1). It is not clarified in [:3.5].
Another definition was suggested .by Seidenberg, Jerard and Magewick [N]. They
sa!. a surface is fair if:
The change In surface currature and in the tangrnt plane is contznuou,c. ((''
continuity): /'
CHAPTER I . AN-INTROD IlCTION TO SCTRFACE ANALYSIS
. there is no unwanted surface inflections:
0 the surface curaature varies in an even manner. I
T h e difference between these two suggestions is tha t it is clearly claimed in [34] that
planar curves should be treated as surface curves.
A surface inflection occurs a t a point where the surface crosses its tangent plane
[1.5] (referred in [34]). We were not able t o find this reference. In our own method.
we use parabolic points instead of surface inflections.
There are two kinds of work dealing with surface fairness: surface analysis and
surface synthesis. In surface analysis, potential design errors are detected, while in
surface synthesis, they are precluded. The similarity is tha t they both have criteria for
surface fairness. T h e difference between them is that the first one uses some method
to detect which part of the surface does not satisfy the criteria, while the second one
constructs surfaces using some constraints based upon the criteria. We will focus 011
the first method in this thesis. 6
1 .Z.3 Surface Analysis Met hods
It is not surprising tha t many different kinds of surface analysis methods have bcen
developed. Previous work focused o s u s i n g shading methods to visualize objects ['I. SOW most of the methods concentrate on using color t o visualize geometric properties
of surfaces or different kinds of characteristic curves of surfaces [6. 17. 191.
Shaded-image rendering
High-resolution shaded raster images can provide concrete visualizations of computer-
generated surfaces. This method is extremely useful when features such as shadowi~ ig~
specular reflection, and depth cueing are included and when the user is free to ma-
nipulate the viewpoint and the positioning and intensity of the light sources.
The basic task in generating high-resolution raster images is t o compute the inter-
section of a set of rays from the \.iewer's exe with the surface. Readers can find morv
about ray-tracing and other rendering methods in [ I%] .
CHAPTER 1. AN 1NTRODlICTlON T O SURFACE ANALYSlS
..
One advantage of direct ray-tracing methods is their flexibility in a110
light sources and shadowing. The determination of the precise surface
ray-surface intersections provides an appropriate shading level for each pixel.
While the generation of shaded images of curved surfaces using rendering tech-
niques such as ray tracing can be a valuablebtechnique for examining surfaces, it is not
adequate to detect all significant anomalies of curvature. It is clearly not practical
for the user to examine surfaces from all possible viewing positions and directions. in
combination with all possible positions of light source(s). It is posible for significant *
- deviation in the curvature of surfaces being modeled to go undetected. This is main1~-
due to aliasing problems [4].
Contour lines
One simple kind of characteristic curve used for detecting surface irregularities is the
contour line. A family of contour lines are defined by a normalized direction vector
r in 3 0 space. The intersections of all planes which are perpendicular to r with thc
surface yield a family of curves on the surface-the contour lines.
Local maxima and minimaof a surface with respect t o the given reference direction
are encircled by closed contour lines, while saddle poirits appear as "passes". In the
exceptional case of a contour at the precise level of a saddle point. the contour lines ,
cross. More complex singular point morphologies on contour lines may also occur'
under exceptional circumstances for higher order surfaces. Nackman [25] presents a
systematic scheme for describing the distribution of such critical points on a surfacc. ?
Diverse methods exist for producing surface,contour maps; they may depend on
a specific surface formulation or apply to general surfaces [B. 331. Plane-sectioning
schemes for parametric poljmomial patches are described by Hoitsma and Roche [16]
and Lee and Fredericks ['O]. P For contouring applications, guaranteeing the correct topology for the sect ion is
crucial. The planar section of a parametric polynomial patch can be described pre-
cisely as a high-order algebraic curve in the parameter space of the patch. and thc* /-=
most reliable met hods of computing the section ' are based on a detailed algebraic
CHAPTER 1. AN INTRODUCTION TO SURFACE ANALYSIS 6
Figure 1.1: Reflection lines: y' on s is the mirror image of the straight line y
analysis of this curve [ l . 111.
While contour lines can be used to detect some properties of surfaces, this method
has its limitation. First, detecting saddle points is dependent on the given reference
direction. Secol~dly. it IS ~ i o t practical to examine all the reference directions.
Reflection lines
In 117. 1 q . 1.cflvctio11 linw are used to detect surface irregularities
Give11 a surface s. an eye point e. a plane arid a family of parallel straight lines ill
the plarie. reflection lines on surface x are the mirror image of the family of straight
lines in tlir plane when looking fro111 the eye point e (see Figure 1.1).
The definition of reflection lines depends on a particular configuratzon. This cori-
figuration co~itains the location of the eye point. the light plane and the direction of
the lines in the light pla11~. Figures 1.2 and 1.3 show reflection lines 011 two diffcre~~t
turbine blades.
The reflection line ~iiethod deterniines unwa~ited curvature regions by irregularitit>s
(see Figurt' 1 3) i l l t l i t ' rt,floc.t io~i-linr' pat t t ~ i of parallel light liries. The disadva~itagv
CHAPTER 1. AN INTRODUCTION TO SURFACE ANALYSIS I
Figure 1.2: Reflection line analysis 1. (Courtesy of Hagen)
Figure 1.3: Reflection line analysis 2. (Courtesy of Hagen)
CHAPTER 1. AN INTRODUCTION TO SURFACE ANALYSIS 8
of this method is that it is not practical t o examine all possible configurations.
Isophotes
Poeschl [30] uses isophotes to detect surface irregularities.
If r(u', v ) is* the parameterization of a ~ u r f a c e and L is the direction of a beam of 4f light, then the isophote condition is
N(u, r ) . L = constant .
where N(u, 2 ; ) is the normal vector of surface r ( u , v ) .
If the surface is Cr-continuous, then the isophotes are Cr-'-continuous curves.
Figures 1.7 to 1.10 show surfaces that do not seem to contain visual irregularities.
The gaps in the isophotes of Figure 1.7 (and also the shading algorithm) prove that
the surfaces are only C'o-continuous, whereas in Figure 1.8 the discontinuities in thc
isophotes show the C1-continuity of the surface at least in isophote directions. The
continuous iosphotes in Figures 1.9 and 1.10 mark, to some extent, C2-continuous
surfaces.
This method suffers some severe drawbacks: 7
Sometimes an obserzrcr with an ill-conditioned line of sight may not clearly r f r -
o g n i x properties of thc isophotes. Either the surface must be rotated or thc
obsercntion point changed.
IVe can test 1,000 isophott dirrctions, but there still can be n gap in t h f ncst
direction.
:Yo one likes to sit i r z front of a graphics terminal for a long t ime and uwtch
isophotes coming up. d b
X method for automatically processing this test is suggested in (311.
CHAPTER 1. AN INTRODUCTION TO SURFACE ANALYSIS
4 Figure 1.4: A surface (in the ~niddle) with two focal surfaces. (Courtesy of Ha,gen,)
Focal surfaces
In 1131. focal surfaces (see Figure 1 .4 ) have been used to detect undesired curvaturcl
situations 011 a surface Hilbert and Cohn-Vossen [14] have a helpful description of
focal surfaces.
Surfaces are parametrically represented as vector-valued functions r ( u , v ) . A focal
surface F(u. 11) is defined as:
where N(u . 1:) is the ~ i o r n ~ a l vector and ti, is ti,,,, or ti,,, (see definitions in Section
2.2.6).
If we consider f u n d a ~ i i e ~ ~ t a l fact,s fro111 differential geometry. it is obvious that the
centers of c,urvat,ure of the ~ior~r ia l section curves a t a particular point on a surface c
fill out a certain seglnerit of the nornial vect,or at this point,. The extremities of thew
segments arfl t 1 1 ~ c ~ n t ~ r s of ciirvature of two principle directions. We call these t,wo
points t hc foml poin t .s of this nornial. f
Hagt.11 [13] iii troduc~s a generation of t,he classical" focal surface concept to
CHAPTER 1. AN INTRODUCTION TO SURFACE ANALYSIS
Figurr) 1.5: A car Iiood with generalized focal surfaces. (Courtesy of Hagen)
achieve a surfacc arialysis tool: .T
where 2 is a real scaling factor to be adjusted for each individual graphics system.
If there are flat points or nearly flat points on a surface. the focal surface pinpoirits
them as shown in Figurc 1.5.
Curvature
Different.ia1 gcionietry dcals with local curvature of surfaces, and in particular shows
that the (local) liat urr of surfaces can be characterized by their mean and Gaussia~i
curvatures. Dill [GI suggests using different colors to represent different values of'
Gaussian curvature. mean curvature. and principle curvatures.
Many niethods have proposed, to use curvat,ure or the combination of color and
curvature to visualize surfaces. In [31] is suggested that pertinent' curvature iri-
forniation be best represented by having a color change represent a percent. cha~igc
in curvature a logarithniic color scale. Elber and Cohen [9] develop a robust, 4
mtthod using hybrid syiiibolic arid numeric operators to creat,e t r i ~ n m d surfac~s .
CHAPTER I . AN INTRODUCTION TO SURFACE ANALYSIS
Figure 1.6: Trimmed surface.
each of which is solely convex, concave, or saddle and partitions original surfaces (see
Figure 1.6). Chapman and Dill [4] suggest using saturation to distinguish bumps and
hollows while still showing the overall variation in Gaussian curvature over the sur-
face, and dso point out that the sign and monotonicity constraints may not always
produce acceptable surfaces-a more restrictive requirement is that the curvature be
convex (i.e. KK" > 0).
1.2.4 Surface Generat ion Met hods
Now we move from surface analysis to surface synthesis.
Network of isoparametric curves
In [lo], the use of curvature plots for the design of curves that have to meet aesthetic
requirements is demonstrated. An algorithm for fixing slope discontinuity of curvature
to fair curves is developed. The algorithm for generating fair curves is generalized
by tensor product method to generate fair surfaces. To fair a surface, first interpret
all rows of the control net as B-spline control polygons and then apply the curve-
fairing algorithm to each of them. In the second step, interpret all columns of the
resulting control net as B-spline polygons and apply the curve-fairing algorithm to
each of them. The final control net will correspond to a surface that is fairer than the
original one.
CHAPTER 1 . AN INTRODl~CTION TO SURFACE ANALYSIS 1'2
In [21], the curvature distiibution of the network of parametric curves is examir~eti
t o create fair parametric surfaces. This technique is based on automatic repositioning
of the surface control points by a constrained minimization algorithm. T h e objective
function is based on a measure of the surface curvature, and the constraint is a measure
of the distance between the original and the modified surfaces. c
Fairness functionals b
Work on the fairness of c u r d s has traditionally focused on the minimization of strain
energy or the arc length integral of the squared magnitude of curvature (see Equation
1.1) ['22].
Traditional work on the fairness of surfaces also focuses on strain energy, minimizing
the area integral of t he sum of the principal curvatures squared (see Equation 1.2)
[26] . l. A
Rando and Roulier [32] propose several specialized geometrically based fairness furlc- - tionals. These functionals are referred as flattening, rounding, -and rolling.
The flattening functional indicates that the minimization of the surface area pro-
duces a faired surface by penalizing some of the conditions tha t detract from a surface's
fairness: large Gaussian curvature. extreme changes in curvature along the lines of
curvature. and large values of principle curvature. This metric reaches an ahsolutv
minimum when K is equal t o 0 everywhere, i.e. fairing with respect t o this metric
tends towards developable surfaces, especially planes. T h e rounding metric penalizes
unwanted flat regions and undulations in the surface. It minimizes the surface area in
terms of the principal curvatures and the ratei'of change of the principal curvatrrr~s
in the principal directions. It has its absolute minimum when ti,,, is equal to K ~ I J Z I J
ever>,where: i.e. its tendencj. is toivards a spherical shape if possible. T h e rolling
functional shows the results of minimizing the surface area in terms of the rnearl arid
CHAPTER 1. AN INTRODUCTION TO SURFACE ANALYSIS
Gaussian curvatures, the principal curvatures and the rates of change of the mean and
Gaussian curvatures along the lines of curvature. This metric enforces yet another
shape-adjustment .- tendency-to produce a cylindrical or conical shape i f possible.
In [24], Moreton and Siquin suggest the following for a curve fairness functional:
This integral evaluates to zero for circular arcs and straight lines.
The corresponding surface fairness functional is
This integral evaluates to zero for cyclides: spheres, cones, cylinders, tori and planes.
1.3 Our Surface Analysis Method.
l lost methods we have reviewed (except those in Section 1.2.4) utilize only the secoricl
derivative of surfaces. In this thesis. we use Bezier and B-spline parametric surfaces to
compute the first and second derivatives of normal curvature and Gaussian curvature
for three special kinds of characteristic curves: planar curves, isopararnetric curves arid
lines of curvature. Based on the sign information of the first and second derivatives
of normal curvature and Gaussian curvature. we can compute fairness measure for a
particular curve network (only for networks of isoparametric curves). We also have
se\.eral methods to visualize the fairness of parametric surfaces. Moreover. we can
detect some potential design errors.
In the following pages of this thesis. first we introduce related background informa-
tion on differential geometry in Chapter 2. In C'hapter :3 , we propose our own method
for computing the first and second derivatives of normal and Gaussian curvatures.
In Chapter 4 a heuristic formula is proposed to compute a quantitative measure of
the fairness of parametric surfaces and several experiments we have carried out are
analyzed. >lore discussions about these experiments are given in Chapter Ti. Finall!,.
we present our conclusion5 in C'hapter 6.
CHAPTER 1 . AN INTRODUCTION TO SURFACE ANALYSIS
Figure 1.7: Co-continuous sur- faces. (Courtesy of Hagen)
Figure 1.8: C1-continuous sur- faces. (Courtesy of Hagen)
Figure 1.9: C2-continuous sur- faces. (Courtesy of Hagen)
Figure 1.10: Color map of isophotes on C3-continuous sur- faces. (Courtesy of Hagen)
Chapter 2
Differential Geometry
In this chapter. wc introduce related different,ial geometry knowledge of curves arid
hurfaces [8. 271.
,'
2.1 Curves
2.1.1 Space Curves
.4 spaw curvv is o t ~ t a i ~ ~ t d wheri a point moves in three-dimensional spacc. call~cl by
~liatlic~ii:itic~ia~is E3 spa(.tl. wh('rr1 E stands for Euclidean.
.A planar c.11ri.~ i h ;t sp;\ctl c.urvc that lies in E2 spacc. i.e. any single fixcd plant..
Figure 2 1. Positiori vttaors
CHAPTER 2. DIFFERENTIAL GEOMETRY
*
Figure 2.2: The chord vector PQ
In intrinsic coordinates. the arc length s is the independent variable, and points
o ~ i thc curve art) described in ternis of their arc length from a starting point s = 0.
The position v ~ c t o r of the starting point is denoted by r (0 ) . and the position vector
of a point that is a distance s along the curve from the starting point is denotcd by
r(s1. These pv i t ion vectors are measured from a fixed origin 0 (3ee Figure 2.1).
The tangent vector to the curve. defined as the derivative of the position vector f
with respect to arc length s. is,a unit vector (Figure 2.2 shows the proof):
ti.) = r t ( , s ) = linl { [ r ( s + A s ) - r ( s ) ] / A s } A 3 4 0
\ZP now look at t l i ~ dt>rivative of the tangent vector and show how it leads to a
dtfinitio~i of curvature Tlit~ derivative of t ( s ) with respect to s is defined by
t'i,,) = lini {[ t (s + A s ) - t ( s ) ] / A s ) 1 , q - U
Sinw t ' ( 3 j is tho clt.rivativt> of a unit vector. it is orthogonal to t ( s ) . It,s ~ ~ ~ i a g ~ ~ i t , ~ i d ( ~
will r l i .p~~id on l i o ~ rapitll~. thr. curve is hending. If the curve is straight at. t,he point
i l l quc.stiori. its ~l iag~ii t l l ( l t~ will be zero. but if the curve is not straight,,.we may writtl
where n ( s ) is a u~i i t vector in the direction of t f ( s ) and K ( S ) is the magnitude of t ' ( s )
CHAPTER 2. DIFFERENTIAL GEOMETRY
Normal Plane \
Figure 2.3: Space curve frame.
K(S) is called the curvature of the curve, and n(s) is called the normal vector at the
point defined by s.
There is an infinity of vectors that are orthogonal to the tangent vector, lying in
the normal plane. Another such unit normal vector that is especially chosen is the
binomal vector b(s), which is orthogonal to both n(s) and t(s) . We define
The three unit vectors t(s), n(s) and b(s) form the space curve frame f (s) (see Figure
2.3):
Points on the curve at which the curvature ~ ( s ) is zero are called points of inflection.
By Equation 2.1 the curvature ~ ( s ) is always positive because n(s) is always in the
same direction as t l(s) (see Figure 2.4). Now we define n(s), arbitrary, to be in the
same direction as tl(s) (or in the reverse direction) at some key point and thereafter
maintain n(s) through sign changes at points of inflection (see Figure 2.5).
The three relations
C H A P T E R 2. DIFFERENTIAL GEOMETRY
K
Points of inflexhn on the parent Curve
S
Figure 2.4: A curvature profile ~ ( s ) if ~ ( s ) is always positive.
Points of inflexion
4' \I v *
S
C H A P T E R 2. DIFFERENTIAL GEOMETRY
are collectjvely known as the Frenet-Serret relations. T is called torsion. which
measures how a space curve is twisting as a point moves along it.
T h e temptationTo organize them into a single matrix equation is very strong:
where
and
2.1.2 Surface Curves
.4 surface curve is a space curve tha t lies on a surface. Whereas a t a point 011 a /+
curve there is one unit tangent vector t. and an infinity of normal vectors forming
the normal plane orthogonal to the tangent vector, the converse is t rue of a surface.
At a point on a surface there is one unit norrnal vector N and an infinity of tangent
vectors forming the tangent plane orthogonal to the normal vector.
At a point on a surface curve. whose arc length meas~ired from a da tum point on
the curve is s . the tangent vector t ( s ) is the same as that of a space curve. T h e changc
in the description of the curve is concerned solely with the normal vectors. The curve c,
normal n ( s ) can now take a secondary role and be supplanted by the surface normal
N ( s ) . The third vector that makes up the surface curve frame lies in the normal p la t~f>
of the curve. and in the tangent plane of T(s), which suggests tha t it is the line of
intersection of these two planes.
CHAPTER 2. DIFFERENTIAL GEOMETRY
Figuro 2.6: The surface curve frame.
T11v surfaw curve f r a n c and it,s relation to the surface are shown in Figure 2.G
Tlic surfac,v frnlnc is illustrattd in Figure 2.7. The connection between thc sl1rfa.c.t.
curve frame F(s) and tlic space curve frame f(s) is
~ ( s ) is the angle of rotation tha t must be applied t o the space curve frame f(s) to
co~iveri it .to the surface m r r r f r a ~ n e F ( s ) It is therefore the angle between b(s) and
N( s ) . and it is also the angle beltween n(s) and T(s) .
CHAPTER 2. DIFFERENTIAL GEOMETRY
Figlirt 2.7: The angle 4.
T h e surfarc curve franie rate equation analogous t o the space curve frame rat,c
Equation 2.2 is given as:
where
Writ i ~ i g
t i y is called the geodeslc curuature of the curve. K , is called the normal curvature of
the curve. and i t is also the normal curvature of the surface in the direction of thc
C H A P T E R 2. DIFFERENTIAL G E O M E T R Y * I , ) --
n curve. t is called the geodesic torsion of the curve, and it is also the tor Ion of the Y surface in t he direction of the curve. \
All curves lying on the surface tha t pass through a particular point in the same
direction must have the same value of K , and t , but they a re free t o have their own
value for K, .
Let us write out the individual equations contained in Equation 2.3 +.
These are known as the Bonnet-Kovalevski relations. They perform the same
function for surface curves tha t the Frenet-Serret relations do for space curves.
2.2 Surfaces
2.2.1 The Equation o f a Surface
In order t o analjvze the properties of a surface at a point, we use the following math-
ematical model for the surface:
where 11 and t' are independent parameters, each varying smoothly over a specified
range of values. i . j and k are unit vectors along the s, y and z axes respectively i r ~
" Cartesian coordinates.
2.2.2 Surface Normal
\\k define the unit surface normal N(u. ( 1 ) a t the point r ( u , r l ) as
CHAPTER 2. DlFFERENTI '4 L G E O M E T R Y
where
2.2.3 The First Fundamental Form
A small excursion .in the tangent plane from a point r ( u , v ) on a surface may he
described as a vector dr:
dr = rudu + r , d u *
dr . dr is called the f irst f u n d a m e n t a l f o r m of the surface:
dr . dr = ~ d u ~ + 2Fdudv + Gdr12
where
The first fundamental form represents the squared magnitude of a small excursion in
the tangent plane from a point 'on a surface.
2.2.4 he Second Fundamental Form
Sow we investigate the movement dN of the tip of N when we move a distance dr
across the surface in a direction specified by choosing particular values of du and dl.:
Because N is norrnal to an!. vector in the tangent plane at a point, we have
N . r , = 0 , .
N . r, . = 0
C H A P T E R 2. DIFFERENTIAL GEOMETRY 24
Differentiating Equations 2.1 and 2.5 partially with respect t o a and 11 in turn. we
get
N . r , , + N u .r , = 0 ('2.6)
N . r,., + N,, . r, = 0 ( 2 . q
N . r,, + Nu . r , = 0 (2 .8 )
N . r ,,,, + N , . r , , = 0 (2 .9)
Because r,, = r,,, from Equations 2.7 and 2.8 we get
From equations 2.6 to 2.9. introducing three more letters, we get
L = N . r,, = -Nu .'r,
J = N . r , , = - N , . r ,
1' = N . r , , , = - N , . r , ,
The scalar quantity dN . dr is called the second fundamental form of the surface:
The second fundamental form represents the projection on the normal plane of a small
excursion in the tangent plane from a point on a surface.
2.2.5 Surface Normal Curvature
One of Bonnet-Iiovalevski relations is
N) = -tint - tT (2 .10)
\\'e dot Equation 2.10 ivi th t . then get the surface normal curvature K,:
K , = ( L h 2 + 2.Uh + L Y ) / ( ~ h 2 + '2Fh + G') (2.11)
where h is a general direction which is represented by du /dr ' . The geometric meaning
of K , is the projection o f t ' on N.
CHAPTER 2. DIFFERERTIAL GEOMETRY
B
2.2.6 Gaussian Curvature
Following 161, we recast Equation 2.1 1 as an ex@eme problem with the implicit
function H ( K , ( ~ ) , h ) = 0, we see tP
Conditions for an extremum are H(h-,(h). h ) = 0, and a H / t l h = 0. The second of
these yields
which, when substituted into the first. gives
K : ( F ~ - E G ) + K,(LG' - 2 M F + I Y E ) + h12 - LN = 0 (2 .13)
This is a quadratic equation with variable ti,. I t is known that for a quadratic equation
ax2 + bx + c = 0. the roots are:
It is also known that
Sow for Equation 2.13. ive have extrenia ti,,;and K,,,. Rut first we get
K , is called mean ( a r t rag€) curraturc and K G is called C;aussian rurra tur t .
Letting
the rnaxirnunl and niinimurn (principal) curvatures are
C H A P T E R 2. DIFFERENTIAL GEOMETRY
Surface shape at a point ca-n be dharacterized by Gaussian curvature (see Figures
2.8 to 2.10. in which the sniall circles are focal points-defined in Focal surfaces.
Section 1 2.3.). v /
Now we look at three types of surface curve with special properties. s
0 Geodesic. lines: If t i , (s ) is zero throughout its length. the surface curve is called
geodesic ltrre.
Asyniptotic lines: If ~ , ( s ) is zero throughout it,s length, the surface curve. is
called asymptot ic 11nr.
0 Lines of curvature: If t ( s ) is zero throughout its length. the surface curve is called
11nc of curraturr which follows rnaxirnunl or mininiuni principle directions.
Figure 2.8: Elliptical Figure 2.9: Hyperboli- Figure 2.10: ~ylir idrical surface. K G > 0. cal surface. tic < 0. surface. KG = 0.
Chapter 3
Derivatives of Normal and
Gau-ssian Curvatures
I n this. chapter. we first summarize the definitions of planar curve fairness and surfacr~
fairness, then we derive our formulae t t compute the first and second derivatives of
normal and Gaussian curvatures. Except as noted. Section 3.3 represents original
*opk and forms a significant part of the contribution of this thesis.
- 3.1 Planar Curve Fairness
Planar curve fairness has been extensively studied [3; 6. :35]. Here we summarize all
these definitions as follows:
.4 planar curve is called f a t i - i f the follo~vi~ig conditions are satisfied:
1. The curve is of CV2 continuit>,:
2. the curve contains a relati\,ely small number of regions which have constant sign 2 of K. djh-[Ids, and 8l log/ t i l l /c is .
)lore specifically. the second corlditiorl means:
2.a The sign.of K charlges as fcw times as possible;
2.b the sign of d l ~ l l d s changes as few times as possible;
CHAPTER 3. DERIVATIVES OF NORMAL AND GAUSSIAN CURVATURES 28
Figure 3.1: Two curves (1) .
2.c thr~ sign of d211~!lltilj/d.s2 changes as few t-irnes as possible.
S o w wt) c'xplaili all t h ~ s ~ (-011ditio1is one by one. if
F i g ~ i r ~ 3.2: Two curvcs ( 2 )
o f a pc~ilit oli tht. curve. i t reflects tlic local property of the curve. Figure 3.1 shows
C , ' o ~ ~ c l i t i o ~ i 2.a st I-ivt,s to 111akt3 tho n u n ~ h r of i~iflection points as s~na l l as possit)l(l.
111 F i g ~ ~ r t ~ :1 2 . to o l ~ t ' i . t111' j)rir~(.lplc of .slrnple~t , s h ( i p ~ , w t should say that thc c3rirvcb
i v ~ t 11 O I I I , ~ l ~ f i t ~ t i o 1 1 poilit i f< i i~ - t l r t 11ii11 tht) otlltr which has many i~~flect , io~i p'oi~~ts.
CHAPTER 3. DERIVATIVES OF NORMAL AND GAUSSIAN CURVATURES 29
Figure 3.3: Two curves (3 )
C'oriclitio~i 2.b statr3s that the. 11u1nber of extreme points of curvature should bt>
as s~iiall as possit)ltb. I11 Figr~rrx 3.3 . thtrc is only one extreme point of curvaturt. 011
( urvt. (L (po i i~ t id to t)y a11 arrO'w). For curve b. there arc thrce extreme points. C ~ ~ r v t >
h fails t o hc fair accordi~ig to condition 2.b. / The usrJ of condition 2.c is based on t,he idea that our eye may be sensitive not
only to curvature and c h a n p of curvature. but in fact to the convexity of curvaturtb
.7i: . . i , ~ . t o ct'1h.l jid.s2. or ~ i i o r ~ generally d2 j ( ~ ) / d s ' . One for~nula suggested by carl i t~~.
ivork i11 ; i l l t O l l l O t ) i l t ' clrbsign [ r ] is f ( K ) 7 1loglh-lI.
For surfac.~ fair11t.s~. O I I ~ ~ suggwti011 is niade in 1351. By intersecting an arbjtrarv pIa~it>
wit11 tht' s ~ ~ r f a c t ~ . ivtl ohtai11 a planar curvc. I f any such curve is fair, then the s u r f i c ~ ~ is
fair. .411otht~r suggvstiol~ i h 111aclc in i10. 311. If thc n ~ t w o r k of isoparanletric curves is
fair. t l i t v : th t> surfaw is fair. Fro111 thcst two suggestions. we have two ways to ex;tmir~t>
hl~rfaccs. 0 1 1 t l is t o t'val11att1 t l ~ c pla~iar clirws. and the other one is to cvaluat,r. tht>
~ivtwork of isol~ara~ilr~t r ic cS11~vcs.
3.3 Implementation Issue
We focus on parametric surfaces. In order t o determine whether a planar or parametric
curve is fair or not. we should be able to compute K , , d ~ , / d s and 8 ~ , / d s ' . We will
derive all of them in this section.
3.3.1 Computing the First and Second Derivatives of Normal
Curvature
The parametric surface r(u. 1 , ) is given by
The surface normal unit vector N is given by
The coefficients E. F. and C; of the first fundamental form are given by
The coefficients L . .\I. and .\- of the second fundamental form are given b ~ .
I 1. = N . r,, = - N u . r, (3.6)
.\I = N . r , , = - N , . r , . = - N , . . r , (3 .7 )
\ = N . r t 2 . = - N , . . r , . (3.8)
The above equations can be found in [',,TI
Computing the normal curvature
I n ['TI. the surface normal curi.ature 6, is given by
K~ = ( L d u ' + L).\ldudt, + . ~ d t l ~ ) / d s '
CHAPTER 3. DER1\/.4TI\'ES OF iVORMA L AND GA IlSSL4N CIIR\'ATIrRES31 I
Computing the first.derivative of normal curvature
From Equation 3.9. we can get t i , , . the first derivative of normal curvature with . I
respect to arc length s:
We need to compute the first partial derivat,ive of L , h!, and :V in Equation 3.11.
From Equations i3.6 to 3.8. we get:
The above equations contain Nu and N,.. Because N is a unit vector, N, and N,, are
contained in the tangent plane. Following [XI. we have
I \vher$;l. B. C. and D are scalars. \\b write the above two equations in matrix form
Sow in order to get '-1. B. C ' . and I ) . ~ v t . nwd
According to Equations 3.6 to 3.8. the above equation can be rewritten as
then we get:
NOW we introduce symbol H :
H = EC; - F 2
\i7e get A , B. C'. and D as follows:
4 = - ( L C ; - l \ l F ) / H
By now. we can compute the first d c r i z ~ a t i d of normal curtqaturt in arq direction h~
specifying d u and d r using the above equations.
Computing the second derivative of normal drvature A
.1 From Equation 3 . 1 1. we get ti,,,?. the second derivative of normal curvature w i t h
respect to arc length s :
The above equation contains the second partial derivative of L . .\I and :V. From
Equations 3.12 to 3.17. ~ v e gt t :
.Iuu = N , , . r u L , + 2 N , . r ,,,, + N . r ,,,, * = N ,.,, . r,,, + 2N,. . r,,, + N . rUL,,,,
. , = N,.,. . r,,. + 2N,. . r ,,.,. + N . r,,,,,,
The above equations contain the second partial derivative of the surface normal vector
N. From Equations i3.18 and :3.19. we have:
Me know that Nu,, is equal to N,, . Sow we need to compute the first partial delitat ive
of '4. B. C'. and D. From Equations 3.21 to 3.24, we get:
'4, =
'4 ,, =
B, =
B, =
(',, =
(~',, =
D, =
D,. =
C'HA PTER 3. DERlV.TI\'ES OF SORhlA L AND G"4 lTSSIAhT ClrRVATITRES:3-I
T h e above equations contain the first partial derivative of E, F, and G. From Equa:
tions 3.3 t o 3 .5 , we get:
From Equation 3.20. we get the final step we need:
H , = E,G+ EC;,, - 2 F f ; l
H , = E,G + EG',, - 2 F F,
By now, we can compute t h e sccond dtrltvatzm of normal curraturc in any direction
by specifying du and dl . using the above equations.
One important geometric property of surfaces is Gaussian curvature. Clic wai t A
to examine the change of Gaussian curvature in ord'kr to get new information for a
surface. Next is the derivation of how to compute the first and second derivatives of
Gaussian curvature with respect to arc length.
3.3.2 Computing the First and Second Derivatives of Gaus-
sian Curvature
Computing Gaussian curvature
Let
Q = rU x r,
C H A P T E R 3. DERIK4TlC'ES OF NORMAL A N D GAliSSIAN ClJRVATlIRES35
Gaussian curvature KG is given by
T h e derivation of t h e above equation is in [6]. Let
Equation 3.29 can be writ ten as:
Computing the first derivative of Gaussian curvature
From Equation 3.31. at3 get KG. . t h e first derivative of Gaussian Eurvature with
respect t o arc length s:
.tic, = dtiG/ds
= [(.Y,Z + .YZu - Z l v \ , L ) Pdu - ( S Z - I " ~ ) P , ~ u
+(.Y,Z + XZ,. - '2\"\.;.)Pdv - (.YZ - \ ' 2 ) ~ c d ~ ~ ] / ( ~ L d s ) (3 .32)
+ c
This equation contains the #st partial derivatives of -Y, 1': Z and P. From Equations
:3.26 t o 3.2s. we get:
From Equation 3.30. we get:
C H A P T E R 3. DERII.:4TI\,-ES OF N O R M A L AND GAllSSIAN ~ ~ ~ R V A T L ~ R E S ~ ~
P, = z(Q Q)(Q, Q + Q . Qv) = J ( Q . & I ( & - Q J (3.10)
Now we need the first partial derivative of Q. From Equation 3.25, we get:
By now, we can compute the first dcrivafive of G'awsian currature in any direction
by specifying du and d v using the equations.
Computing the second derivative of Gaussian curvature
We let
1. = (XZ - Y 2 ) P 1 , %
J then from Equation :3.:32. 1r.c get a simpler for~n:
From Equation 3.45, we get tic,,, the second derivative of Gaussian curvature with
respect to arc length s:
The above equation contains the first partial derivative of S, T. 1' and 1.. Me need
the following equat ion:
CHAPTER 3. DERIKATIL'ES OF NORMAL AND GA USSIAN CI!RI/'ATZrRES S'7
I;. = (.u,Z + .YZ,. - 'L1TJP,, + ( S Z - Y2)PL,L, ( 3 3 1 )
\.l:e also need to compute t tic second partial derivat ive of .Y. Y* and Z . From Equations
3.33 to 3.38. we get:
CHAPTER 3. DERIVATIVES OF NORMAL AND GAIISSIAN CURVATURES38
Equations 3.44 to :3..51 co1itai11 t h e second partial derivative of P. From Equations
13.39 and 3.40, we get:
CHAPTER 3. DERI\.ATI\.'ES OF XORhl,4L A N D G A U SIAN ClrRVATliRES39 4
'd So\;. from equation 3.41 and 3.4'2. the final step is as follows:
Thus. we can now compute the sccond deriziatire of Gauss ian curva ture in any direc-
tion by specifying dtl and dl . using the above equations.
3.3.3 Computing Any Order Partial Derivative of Parametric
Surfaces t
In the above derivation. we need to compute the partial derivative of surfaces from
the first to the fourth order. such as r,. r,,. r ,,,, and r ,,,,. It is relatively simple
to compute any order partial deriiative of Bezier and non-rational B-spline surfaces.
The equation of computing a n . order partial derivative of NI'RBS surfaces ca11 be
found in ['29].
3.3.4 Computing the Tangent Vector (du . At)
Based on the above derivation. \ye can choose du and dl9 to control the type of curves.
111 the following we discuss three kinds of characteristic curves: planar curves. isopara-
metric curves and lines of cur~.aturc.
Planar curves
A planar curve is obtained b!. intersecting a reference plane with the parametric
surface. Suppose the reference plane is
CHAPTER 3. DERlVATII'ES OF .VORMAL AND GA I:SSIAiV.CCIRVATIJRES.10
and the parametric surface is described by Equation 3.1, then the corresponding
planar curve is
j ( u , 27) = "h(11. r * ) + By(u, 2 , ) + C z ( u , 21) + D = 0
T h e tangent vector (du,dr . ) at point ( u , tv) is
(du.dr.) = (f,.. f,) 8
Isoparametric curves
Surface shape is usua l l~ . conveyed by the representation of curves of constant pararn-
eters, in particular t he surface patch boundaries tha t occur a t the knot values [21].
Now we consider isoparametric curves in which the parameter u or r7 is constant. The
tangent vector (du, d l ? ) at a point ( u . ( 7 ) of this kind of isopararnetric curve is
( d u . drl) = ( c l . c:!)
where cl and c2 are both co~istants and one and only one of them is zero.
Lines of curvature
Lines of curvature are an important kind of characteristic of surfaces. The tangent
direction of lines of curvature can he ohtairied through Equation 2.12 by substitution
of ti, with A-,,,~ and K ,,,. jl'e can also examine other ki~ids of characteristic curves (reflection lines. geodesic
lines. e tc . ) by computirig their appropriate tangent vectors.
3.3.5 Computing the Fairness of a Network of Isoparametric
Curves
In our method. ice construct t hc ~ietwork of isopararnetric curves using sample points
on the surface by specif!.ing the "resolution" of the surface. T h e resolution of t h e
surface is the number of row.; (c-ollir~ins) of a single patch. In Figure :3.4, the network
of isoparametric curves corlsists of two patches:
CHAPTER 3. DERIVATIVES OF NORMAL AND GAUSSIAN CURVATURES41
Patch 1 Patch 2
Figure 3.4: Surface mesh
I ~ ---- I l l
in. 0 1 11.01 u (0.01 11.0)
f'aiih I Parch ?
I I .11 11.1) to. I I I (0.11:
F i g u r ~ 3.5: Parametric spacc. " .
. I
The. surface rcsolutiori is 1. Now wc can ~valliat,c 11 isoparanletric curves:
J
\\'e can reconstruct the network of isopararnetric curves to get more or less isopara-
metric curves by changing the patch resolution. Figure 3.5 shows the pararrictric
space of the netivork of iwpararnetric curves.
The following is the algorithm of evaluating the fairness of surfaces.
1. Set the resolution of the surface patch;
2. Choose one parametric direction by setting ( d u . d t * ) to he ( 1. 0 ) :
.5. Choose another parametric. direction by setting (du. d t ? ) to be (0 . 1 ) :
the isopararnetric cur\ .vh of corlstarit u :
8. Compute the fairness of the rlc.tivork of isobararnetric curves based on the results
from step 1 and 7 .
3.3.6 Computing Planar Curves and Lines of Curvature
In our implementation. we choose sample points on the surface and interpolate be- - tween them. That is one reason why we cannot examine a planar curve automatically.
\Ve can only present the sign information of one particular family of planar curves to
the user at one time. and let the user check it out.
The same situation is for lines of curvature. We can only present the sign infor- .
mation of lines of curvature to the user at one time. and let the user examine the
result.
3.3.7 Computing the Maximum Change of I d l~ . l /dsI
The way to find the maximum change of Idlti,l/dsl for every sample point is to first
specify the number of directions we want to compute. then compute jdlti,j/ds( for
every ~arnple~direction and take the maximum value. Because two principal directions
are perpendicular to each other in sy: space but not necessarily in parametric spact,.
the distribution of the samplc directions in sy: space must be converted to that i l l
p a rahLr ic space.
3.3.8 Computing the Integral of ld l~,, l/dsl
\ i e can specif the number of directions we want to compute for every sarnple p i ~ i t
and then compute Idltinl/d..i for e\.erj. direction. The normalized total of Idlti,l/d>l
for all directions is the integral of ldl~,~l/d.sl.
- \>
3.3.9 The Interface of Our Program SURF
Figure 3.6 sho\vs that the absolute \ d u e . i.alue and sign of Gaussian. a \wage (mean)
curvature. maximum and mirtinium principdl curvatures. normdl curvature. torsio~i.
and t,he first and' seconcl d e r i \ . a t i \ ~ of normal and Gausgian curvatures car1 he die-
plaj.ed through linear. c-onic ; i r ~ t l csporlcnt ial color maps. Figure 3.6 also sho\vs
ive car1 compute the fairr~v-. o f rrt~t\vork~ of isoparanietric cur\.es ( the isoparar~ir.tric
crlr\.t3s can be chosen from paramet r i c direct ion of constant ( 1 . paranictr ic direction of
c o ~ ~ s t a n t 1 . . or ho th pa ramet r i c directions of constant c i a n d 1 . ) .
F i g ~ ~ r c 3 . 7 shoivs we can c o n i p r ~ t e tiircc kinds of cur\.cs: lines of cliri.aturc. isopara-
~i~c>tr . ic curves and planar curves for t h i r gconnetric properties. \\.& car1 also spc>cify
t ti(. ivcights for coniput ing t hc fairncw of networks of isoparariltxtric curves.
111 this chapter . tvc dr.ri\.(d t he forrnr~lac t o cor l ip~ i t c t IIC> first a n d scconcl derii.ati\.cs
r o 1 t 1 1 1 . \\>. \v i l I a l ~ tlircrlss t I I V user t c ~ t ing \vc 11ai.c carried o r ~ t t o show hotv
CHAPTER 3. DERIVATIVES OF NORMAL AND GAUSSIAN CURVATURES 45
Figure 3.6: The interface of SURF (1).
CHAPTER 3. DERIVATIVES OF NORMAL AND GAUSSIAN CURVATURES46
Figure 3.7: The interface of SURF (2).
t .
Chapter 4
Analysis
In this chapter, we first examine two well-known shapes: the torus and the lid of I'tah
teapot. It is shown that geometric properties of parametric surfaces such as Gaussiari
curvature. normal curvature. t11e first and the second derivatives of Gaussian and nor-
mal curvatures for three kinds of curves (planar curves, isoparametric curves and lines
of curvature) can he inspected using our algorithm. Then one example is discussed
to show that our algorithm i \ effective in detecting potential design problems. We
also describe several experirncnts we have carried out in order to test how effective
our method is in measuring aesthetic qualities of parametric surfaces. One formula is
proposed to compute the fairness based on the number of segments of constant sign
For all the images in this c-l~aptcr green represents zero. purple represents nega- - tive value, red represents posit i1.c value and black represents value smaller than the
smallest value in the legend i f t l~ere is rlo explicit explanation
4.1 Torus
The first example we exaniil~cl i.; the torus.
CHAPTER 4. ANALYSIS
4.1.1 Planar Curves
T h e parallel reference planes are described by the equation: -
whgre c is a constant.
By intersecting the torus with planes described by Equation 4.1, we get planar
curves on it. Figure 4.1 shows the distribution of tier; on torus. Figures 4.2 and 4.:3
show d J ~ ~ l / d s and 62110~ [ t i c / l / d s L of the planar curves, respectively.
Figure 4.4 shows the distribution of K, of the planar curves. Figures 4.5 and 1.6
show d l ~ , l / d s and dZl/oglti, 1 l /d.s2 of the planar curves, respectively.
Figures 4.7 t o 4.12 show the same properties of torus as ~ i ~ u r e s 4.1 to 4.6.
but the sign instead of the \.slue is displayed. Note tha t the small green areas on the
right side in Figures 4.8. 1.9. 4.11 and 4.12 are introduced by the limitation of t h e
surface resolution.
From Figures 4.1 to 4.6 or from Figures 4.7 t o 4.12, it is easy to tell that
the distribution of C;aussia~i curvature is more symmetric than the distribution of
all other properties. while n.11at we cxpect is that the distribution of 6,. dlti , 1ltl.y.
62110glti,l [ I d s 2 . d l t iGl /dh and tlL lloY ltir;l l / d a L has the same degree of synirnetry as
K G . If we use parallel refercncc planes z = c o n s t a n t , the distribution of all the six
properties will have the same tlcgree of s!,rnnietry. Different networks of planar curves
for a surface can be generated t lirough specifying different plane equations. Although 'a
this gives us flexibility t o aiial!.ze surfacc. it is also problematic hecause we do riot
want t o specify different pla~ir, cyuations for different surface.
4.1.2 Isoparamet ric Curves
In Figures 4.13 and 1.14. t l l ~ pasanictric direction of constant 11 arid t1 are sho\v11.
respectively. - Figure 4.1.5 shows k , of t l ~ t ~ isoparametric curves of constant u and Figure 1.16
shows k, of the isoparaniet sic curI.es of constant I , , while dlti , I / d s arid dLI loy [ti,, I l /d .sL
of the isoparametric cus\.cs 0 1 ' conr;tatit ( 1 . and dlti , l /ds and dL l loyJ t i n ( l / d . s2 of thv 6
CHAPTER 4. ANALHIS
I constant 1. / 1.0 / 1.0 I 1 .O 11.0 [ 1.0 1 1 .O I Table 4.1: Average nuniber of segments of constant sign of the torus.
isoparametric curves of constant l 1 are shown in Figures 4.17, 4.19. 4.15 and 4.20
respectively. In theory dlti,, ll(1.s and d2110glti, ll/ds2 for the isoparametric curves of
constant u and rv should bc zcro. however they are not in our computation because of
computational approximat ion.
Figures 4.21 and 4.22 show the same properties of torus as Figures 4.15 and
4.16, hut the sign instead of tile value is displayed. Here we do not give the images
with sign of dlti, I/ds arid d211~,y l~ , 1 l/d,qL for the isoparametric curves of constant 11
and constant z 3 because t h c ~ - ;it.(. all zero.
Figure 4 . 2 3 shows clltic;l/il.~ for t h e isoparametric curves of constant 1 1 , while Figure
4.24 shows d2110g[tiGll/ds2 for the isoparametric curves of constant u. d l ~ ~ l l d s arid
d2110gltiGll/ds2 are all zero for the isoparanietric curves of constant v. Figures 4.25
and 4.26 show the same properties as Figures 3.Z3 and 4.24, but the sign i~istead
of the value is shokvn.
For both parametric direction u and 11. we compute -50 isoparametric curves for
;each of them. .Thc values i 1 1 Tablc 4 . 1 s11ow the average number of segments of
constant sign which are oht airlcd through t hc algorithms in last section.
4.1.3 Lines of Curvature
In the parametrization of the torus. because the parametric direction of constant
u is the same as the n i a s i ~ i l r ~ t ~ ~ principal direction and the parametric direction of
constant z 3 is the same ah t h t . ~ l~in in iuni principal direction. the con~putation for the
isoparametric cur\.es of co~ista~lt 1 1 a!id 1 % is the sanle as that for the lirlcs of curvat rirr3
of maximum and mininilirli pt.itlc.ipal directions. respectively.
CHAPTER 4. AN.4 LI.-SIS
4.1.4 Summary
We know tha t all t he lines of curvature of the torus are actually planar curves. So
are all the isoparametric curves for the parametric direction of c o n a a n t u and t * . .A11
the lines of curvature along the ~naximuni principal direction can be obtained in this
way: suppose there is a straight line perpendicular t o the xy plane and through the
center of the torus, all the planes containing this straight line intersect with the tor~is .
T h e intersections are lines of curvature along the maximum principal direction. :Ill
the lines of curvature along the minimum princifl direction can also be obtained
by intersecting the torus with a family of parallel planes which are perpendicular t o
z-axis.
4.2 Lid of Utah Teapot
The second example is the lid of I ' tah teapot. Me only discuss K , and its derivatives
because K G and its derivatives have similar distribution as ti, and its derivatives have.
r
4.2.1 Planar Curves
Figure -1.27 shows the (list r i l ) ~ ~ t ioli of ti,, of t tic, planar curves tvhich are obt aiiiecl I)!
intersecting the lid with pla11c.s pcrpent l ic~~lar to s-axis. Figures -4.28 and 1.2'3 show
d l ~ ~ ~ l / d s and d2l1oglti,ll/d.,' of the planar curves. respectively.
4.2.2 Isoparametric Curves
In Figures 4.30 and 1.:{1. t h t x para~netr ic direction of constant (1 and t1 are shown.
respectively. Figure 1.:32 slio\is the nornlal curvature of the isoparametric curves of
constant u , and Figure - I . :XJ sl~o\vs the ~lorrnal curvature of the isoparametric curves a of constant r , dl t i , l / d s and d L l l o y l t i , l j /dsZ of the isoparametric curves of constant
u are shown in Figures 4.:31 and 4.136 respectively. Figures 4.:3.5 and 4.37 show
d ( t i , I /ds and d2110yltin 1 I/d.\' of the iwpara~net r ic curves of constant t a , respectivel!..
CHAPTER 4. ANALLSIS
Table -1.2: Analysis result of the lid of Ut,ah teapot.
constant u
constant t 7
From Figures 4.34 and 1.36. wecan tell that thesignofdlti,I/ds and 62110gl~,II/d.s2
of the isopara~netric curves of constant u changes several times. which means the value
of dlti,I/ds and d2110glti,, II/dsL is not zero. But if we have a perfect lid, which is really
a surface of revolution. then dlti, l / ds and d2110glti, ll/ds2 of the isoparametric curves
of constant ZL should be zero. They are not zero in our results because we use Bezier
curves to model circles (i.e. they are approxiniatio~ls).
The resolution of a single patch is 20. and the lid contains 8 patches. Table 4.2
shows the result of the anal~..is.
4.2.3 Lines of Curvature
n
3.47 1.00
In Figures 4.38 and -1 .:3Y. the ~liasimurn and mininlum principal directions are shown.
respecti\rely. Figures 4.10. 4.42 arid 4.44 show K,,,,, d l ~ ~ , ~ l/ds and d2110glti,,,,ll/dsL
of lines of curvature of n ias in i~ r~n principal direction, respectively, a i d Figures 1.11.
4 .43 and 4.45 show ti ,,,,,,. dlti ,,,,,, I/(/.\ arid d2lloy (ti,,,/ [ Ids2 of lines of curvature of
minimum principal direct ior~. ~c,ypect ii-el!..
4.2.4 Summary
d)ri, d s
6.52 2
From the two exan ip l~s abo1.c. I\.(. find that e \ m though the isopararrietric curve is not
an intrinsic property of paraliir~tric surfaces. i t is quite effective to show the geometric
properties of parametric su r fac~ t~~ . \\'c ca11 see that using isopararnetric curves is more
effecti\.e than using l i n e of c.l~r\ature for t l ~ e lid to evaluate the surface. Both torus
and lid are surfaces of rc>\.ol~~t io11. \\c. sliol~lct not ice that all the isoparamet ric c u r \ w
are also planar cur\.cs for t h t . torus ar~tl the lid in their parametrization. The o n l ~ .
d Z I / o g l ~ n u ds2
6.00 5.79
KG
4.00 1.00
ds
6.00 9.26
ds ~ ' I ~ o ~ G U
S.OO 1.00
J
CHAPTER 4. .41V.4L\7S1S
difference between isopararnetric curves and planar curves in these two examples is
the way we choose the reference planes.
Based on our analysis of se\.eral kinds of parametric surfaces, we conclude that
isoparametric curves are effect i\-c characteristic curves for surface analysis. Of course
we need an appropriate paramcterizat ion for parametric surfaces.
In the two examples aI3oi.c. we show that our method can be used in visuall~.
inspecting parametric surfaces through the display of ti,, d l ~ , ( I d s , 61 IlogItinI I / d s 2 , K G .
d l ~ ~ ( / d s arid 8 l l o g l t i ~ l i / d . s ~ . and gives us more geometric information of parametric
surfaces which cannot hv ohtai~lcd through Gouraud shading.
Sow let us look at a real \vorltl ohjcct pat?. part of the outer surface of an automobile
(see Figures 4.46 to 1 .19) .
The direction of the ( 1 parameter is shoiv11 in Figure 4.46, while Figures 4.47.
4.38 and 4.49 show displa~- of the sign of A-,. dlt i , l /ds and d2110glti,ll/ds2 for thc
isoparametric curves of coristalit u.4-ndcsirable '.wri~lkles" on the surface are clearly
evident in the display of tmtli t I r ( 1 first derivative of h-, and the second derivative of
K,. Thus we see that o t ~ r rlic'thocl is effecti\.c in detecting potential design problelns.
4.4 Bump
Sow let us esan~iric a si~lglv 1):1:( 1 1 hurFac~.: l)1111ip (see Figure .4.64). In this exarr~plc.
we only consider K , and its c l t 1 1 . i i a t i\.ch of t Irt, isopara~net ric curves. P
In this example. becauw t l r t , isoparariietric curves of constant u ha\.e the saIiirl
geometric properties as tlrosc of constant - 1 % . ive will only examine one of therri.
Figure 4..50 shows the directio~i of constant u . Figures -1.51. 4.52 and 4.5:3 show
K , . dlt i , l /ds and d2110yi~,, I/tl--'. rcspccti\.cl!.. while Figures 4.54. 4.55 arid 4.56
show the sign of A-,. Cilti. I / ( / . + a l~( l c1L/109~ti,, / I /d.qL. respectively. The resolution of this
single patch surface is 50. l ' l i t , ;ilial~.sis 1.c.sl11t is in Table -1.3.
CHAPTER 4. ANALYSIS
4.5 Improved Bump
We edit the bump to make it smoother a t the four corners (see Figure 4.65).
Figure 4.57 shows the direction of constant u . Figures 4.58, 4.59 and 4.60 show
K , , dlti,l/ds and d2110glti,711/d~L. respectively, while Figures 4.61. 4.62 and 4.63
show the sign of K , . d(~,l/d.\ and c12(loylti,,I(/ds', respectively. The resolution of this
single patch surface is 50. Tlic i~nal>,sis result is in Table 4.4.
We asked 13 people to < c l ( ~ t which surface of these two bumps they prefer. 12
of them were graduate students in computing science and 1 of them was a facult).
member in computing science. I he testing result is in Table 4..5.
Here we suggest a formula to compute the fairness of one isoparanletric curve:
In the equation aho1.e.
W'" (I,%,) is the ivcight for the pararnctric direction of coristant u ( 1 ' ) ;
It; is the weight for hot 11 t ; , , aricl t i c ; :
0 I,t'l is the ~veight for l)ot 11 (llti,, 1 l d . s a ~ ~ t l dltic;l/ds:
!V (0, (.YKco,) is the avcragt, ~iurnber of segrllents of constant sign of ti, for the <nu n c
isoparametric curves of col~starit 1 1 ( 1 % ) ;
0 .\.' , 2 , (.\.' ( 2 , ) is t hc ;1\.t.ragfx r~~irillwr of s c g ~ ~ i e ~ i t s o f cor~stant sign of d'IloyI~,,/I/d..;l n u 'in,
for the isopara~iletric c u r \ c's of col~btaiit 11 ( t , ) :
Lr
,.I! ( o ) (,V ( o ) ) is the average number of segments of constant sign of K G for the '(cu '((3"
isoparametric curves of c o ~ ~ s t a n t u ( c ) : ,'
.WK(l) (.lrKil) ) is the avcragv number of segments of constant sign of d l ~ ~ l / d s for Gu GI.
the isoparametric cur\.cs of constant (1 ( t * ) ;
X ( 2 ) ( ;YK(,) ) is the avcragv numlwr of segments of constant sign of P 1log/ticl '(GU G v
for the isoparametric c u r \ w of constant (1 ( 2 3 ) ;
For a parametric surface. we compute the fairness for every sampled isoparametric
curve, and use the average fair~less as the fairness of the parametric surface. The
smaller the fairness. the het ter t l i t > aest hct ic quality of a surface.
If we use the weights listetl i l l Tahle -4.6. arid t h e average numbers listed in Tables
1.3 and 4.4, we get the fairness of thew two bumps in Table 4.7. which contains also
the average rank ( i f a surface is preferred. then its rank is 1, otherwise its rank is 2 ) .
The fairness of the improved h1111ip is 1.61 while the fairness of the original bump is
'2.38: we conclude the irnpro\.cd hump has better aesthetic quality than the original
bump. based on our rnctllod. E'so~ii Tahlc 4.5. we know that 10/(10 + : 3 ) = 77% of 1 3
subjects thought thc inipro\.tcl 1)unip is 1)cttc.r than the original bump. The average
rank of improi-ed hump is 1.2j i r ~ ~ d t hc average rank of bump is 1.77. which tells us
that the majority of the sr11)jcct. thought that the improved bump is better than the .
original bump. This shoivs that our ~iir,thod is effective in measuring the aesthetic
qualities of parametric sr~sfac-txi i l l this ot,sc~svctl example.
4.6 More Tests
In the follo~virig. ive i v i l l t l r ~ . ~ c r i l , ~ ~ fi\.c ~ ~ i o r t ~ tcsts. lGicli test contains :3 or 1 paranietric
surfaces. For each surface. t l i t b fairness is computed through our method based on
the number of segments of const i t l i t sign of normal curvature, Gaussian curvature and
the first and second deri\.at i1.c. of rlorriial and Gaussian curvatures, while the average
rank is computed fro111 t Ilt , ral~l; 1i.t oht ailled fro111 the subjects.
s'.
4.6.1 Four Saddles
We constructed four saddles (-set, Figures 4.66 to 4 .69) and asked 14 people t o analyze
them, 12 of them were graduate students in computing science and 2 of them were
faculty members in computing science. The surfaces are selected in the order based
on their aesthetic appearance. I ' he earlier the surface is selected, the better aesthetic
quality it has. T h e testing r c s ~ ~ l ! is listed in Table 4 .8 . T h e analysis result using
our method is lis.ted in Table I . ! ) . 111 our canputat ion. the resolution is 2.5 and each
saddle contains 8 patches.
If we use the weights listed i l l Table -4.6. and the average numbers listed in Table
4.9 . we get the fairness of thew four saddles in Tahle 3.10, which also contains the
average rank.
4.6.2 Three Telephone Handsets
In this test and the follo\vir~g to.ts. \ v r askrd 12 subjects ( 1 1 of them were graduatcx
students in computing scic~icc. i111c1 1 of them \vas a faculty nieniber in computing
science),
Three telephone hanclwts arcn sho\vr~ in Figures 4 .70 to 4.7'2. Table - 1 . 1 1 shows
the testing result fro111 t he sut~, ic~. t 5 . .laI)lr -4.12 shows the analysis result f r o ~ n our
method. I'sing the iveight.; i l l 'I<rt,lc 1.6. \vc get the fairness of these three telephonc
handsets in Tahle . 4 . 1 : 3 . ~ ~ 1 1 i c . 1 1 ( o ~ ~ t a i ~ ~ . . also the average rank.
4.6.3 Three Mouse Tops
Three mouse tops are shoivn i r ~ F'igurcs 1.73 to 4.75. Tahle 4.14 shows the testing
result from the subjects. l'ahlt. 1.15 sho\v.; the analysis result from our method. I'sing
the weights in Table -4.6. \\.t. gc.1 t 11e fais~ltlss of tlicsc three mouse tops in Table -1.16.
~ h i c h contains also t hc ;1\.f,r-aqt3 ~ . i t n k .
4.6.4- Four Vases
Four vases are showi in Figures 4.76 to 4.79. Table 4.17 shows the testing result
from the subjects. Tablr -1.1 S sl~o\\-s t hr analysis result from our met hod?--17sing the
weights in Table 4.6. ive gct tlic fairness of these four vases in Table 4.19. which
contains also the a\.eragc rank.
4.6.5 Four Perfume Bottles
Four perfume bottles are show11 in Figurcs -!.SO to -1.S3. Table 4.20 shows the testing
result from the subjects. Tahlc. 1.21 sho\vs the analysis result from our method. Vsing
the weights in Table - 4 . 0 . \vc get t t ~ c fairriess of these four perfume bottles in Table
4.22. which co1_ltairl5 also tht . ai.tlragc rank.
For all our test.;. \ vc uscd ~iriifvrrii \\-eights to compute the fairness of surfaces.
\ \ e also tried using 11on-u11ifor111 \v~ights. for example. larger weights for nornial and
Gaussian curvature5 t ha11 for t l i f , first clvri\.at ives of nornial and Gaussian curvatures.
and larger wigh t s for t h r ' lat t t ~ t 1la11 fos t 11c second derivatives of normal and Gaussiar~ 1
curvatures. \\k fouricl t11at ~ ~ . i r l c , 111 i i f0s l l l \\-eights is more effective than using rion-
uniform weights. Orit. possil,lc \v;i!. t u ahsigl.1 the ~veights might be to get enougll
statistical inforniatiori arrrl t h c ~ n sol\.c for t l i ~ 1 1 1 usirig Equation 4.2.
In this chapter. n.e sho~vcri that our nlettiod can be used to visually inspect geo-
metric properties of pararl~ct ric s~1rfacc.s arid also effcct ive in detecting potential design
problems. One test co~itailri~ir: t \ v o 5usfac.c.s \vas discussed to show that our method
is effective to nieaiurv ac3.t l 1 f . t ic cl~~alit!. of paranictric surface. Then we presented 3
more tests that gi1.c. 11s rl1ot.r. saiv data t l l t t t \vv C ~ I I anakze . In next chapter. we will
discuss how effccti\x. 0111. ~~ic'tliocl i. i l l ~ l l t ~ ~ ~ ~ u s i r i g the aesthetic qualit!. of parametric
surfaces.
1 constant t q 1 1.00 1 2.88 1 1.92 / 2.9.' 1 3.92 1 1 . 6 I Table 1.3: .4\.eragc ~iurnbcr of scg~nents of constant sign for bump.
constant u
Table 1 . 4 : Averagt 11111111)er of segnimts of constant sign for improved bump.
dlh, d s
'2.88 ti n
1 .OO
/ l'rcfered stlrfacc j Surnber of people ]
I I.' ' . ( i i ~ r~css 1 Avcraee Rank 1
d 2 ~ I O ~ ~ K ~ U d s 2
1.9%
-1'ablr~ -4.7: F'air~iess of bunips.
KG
2.9%
u. da
3.92 ,
d z l l o 9 1 ~ G ~ ~ '
da2
1.64
CHAPTER 4. .A$.-\ L1.SI.S
I Outcolile I ?;umber of people I
saddlel, coristarit 1 1
Tal,lt' 1. 1 1 ) : I . ' ; I ~ I . I I ( + \ i 111c1 ;i\.c.ragr. rank for saddles.
,L
1.00 i
saddlel. constant 1 .
saddle2, constant 1 1 - saddle2. constant 1%
saddle3, constant 1 1
i s
2.41 1 .OO
1 .OO 1 .OO
.i.OO
5.00 3.49 3.00
3.6.5
1.00
1.0 1.00
5.00 ,
4.00
2.08 2.00
8.00
d2110gl~nll ds2
2.92 3.30
3.06 2.00
8.00
3.00
2.34 3.00
1.37
d2 d q 2
3.00 K G
1.00 Aa
2.00
1 O u t c o ~ n c I Surnher of people 1
7'at)lc 1 . I 1 : 'l'('l;til~g ~x ' s l~ l t of telephone handsets.
, L , I I I Telephone handset 2. . ori.\tant 1. I 1 0 0 / 2.00 1 3.00 1 1.00 1 2.00 I 3.00 1
, , 1
, I.'airncss Average Rank
2-17
Telephone handset 1. c.or~sta~it
I Telephone handset 2. c o ~ ~ s t a n t rl
1 I
Telephone handsrt : 3 . ( O I I ~ ~ H I I ~ I I 1 5.00 1 S.OO Telephone handst.1 : 3 . c.c,~i,talit 1 % 1 1.00 ! 2.00
Table - I . 1 :i: I - ' , I ! I : i 1 7 - - A\ 1 I C I Z I . ra11k for telephone handsets.
1.00 1.0
1.00
1.00
3.00 2.08
1
3.30
:j.Otj
-5.00 3.49
8.00 2.00
1.37 :1.00
3.00 2.33
3.65 3.91
5.00 1.00
O i i t c o i i S ~lii iber of people 1 . 2. 3 I '>
- -
l'al)lt, 1 . 1 4 : I ' c s t i ~ ~ ; rc~sult of mouse tops.
Xlouie top 2. c o ~ i h f ; r 1 1 t (1
blouse top 2. c'onbtiilit 1 %
hlouse top 3. c . o n c t i l l 1 1 -71
Mouse top 3. coll\t i 1 1 1 1 r ,
1 .OO ' :i.06 71.00 I 2.00 3.00 \.(I0 1 .00 0 0
?.:I4 3.00
1 . :3.00
1.0 1.00
5.00 1 .OO
2.08 2.00
8.00 2.00
3.49 3.00
3.6.5 3.91
CHAPTER 4. A4Y.4L\~.S1.5'
O u t come 1 ~ ~ ~ n b e r ;f people 1 ( 1 . 2. :3. 11
I Vase 1. c o i l s t n n m n i 3.00 i 1.00 i 4.00 i 5.00 1
Vase 2, coilstailt 1 .
Vase 3. constant 11
I t11 , ! t s -1. I ' 1 : I . ' i l i r l ~ c ' \ - 1 1 1 I it\.tv.agc r a n k for i.ases.
1 1
Vase 4. coi~staiit : I 5.00 Vase 1. coi~.tant ,. 1.00
1.00
5.00 Vase 3, constant 1 . , 1 .OO
2.00 / 3.00
S.00 ! 1.:3T
t 2.00 :3.00
h . 0 0 1 . 7 2 . 0 :1.00 L
I
1.00
5.00
5.00 1.00
8.00 2.00
1.00
3.6.5 :3.91
1
2.00
8.00 2.00
3.00
3.65 3.91
CHAPTER 4. .AS.-\ Ll-$1.5
j Outco~lie \'umber of people I
h', , -- i.3
Perfume bott lc 1 . cor1.1~11it 11 2 . 4 1 Perfume bott 11 1 . c o I l ~ : ~ . I O - -
.;-</IillT : ~ : O F
. l i l t 1 . 2.00
S O 0 Perfume bottle : 3 . corl.1 i l l i t ( * 1 . O O 2.00 .-).OO 5.00 1 7 5.00 8.00 3.65
- -- 1 .Or1 2.00 3.00 1.00 2.00 3.91 p~ -
d 2 1 1 ~ g ) ~ n l l As2
2.92 3.00 2.R4 3.00
1 J i 3.00
KG
1.00 1.00
1.0 1.00
5.00 1.00
d .q
2.00 4-00
2.08 2.00
8 0 0 2.00
d ' l l 0 ~ 1 ~ ~ U *
da2
3.00 5.00
3.49 3.00
3.65 -
3.91
CHAPTER 4. ANALYSIS
Figure 4.1: Torus, KG.
Figure 4.2: Torus, dlnG, [Ids. Torus,
Figure 4.4: Torus, n,,. Planar curves are perpendicular to x-axis.
CHAPTER 4 . ANALYSIS
Figure 4.5: Torus, dlrc,, 1 Ids. Figure 4.6: Torus, @11ogl&%, Illds2.
5 . - .. .
Figure 4.8: Torus, sign of dlfiGS 1 / d ~ -
Figure 4.7: Torus, sign of KG.
Figure 4.9: Torus, sign of @ I l o i d w + J l / d ~ ~ .
CHAPTER 4 . ANALYSIS
Figure 4.10: Torus, sign of K,, . Planar curves are perpendicular to x-axis.
Figure 4.11: Torus, sign of dlKn, I p s .
Figure 4.13: Torus, parametric direction of constant u.
Figure 4.14: Torus, parametric direction of constant v .
CHAPTER 4. ANALYSIS
'Figure 4.15: Torus, 4;. Figure 4.16: Torus, K,, .
Figure 4.18: Torus, d l ~ ~ , )/ds.
Figure 4.19: 8Ebl &nu l l /ds3.
Torus, Torus.
CHAPTER 4. ANALYSlS
Figure 4.21: Torus, sign d a,.
Figure 4.25: Torus, sign of ~ I K G , I/ds.
Figure 4.22: Torus, sign of rc,,.
Torus,
Figure 4.26: Torus, sign of @Iloglw, I Ips2.
CHAPTER 4. ANALYSIS
Figure 4.27: Lid, sign of b,. Planar curves are perpendicular to x-axis.
1 Figure 4.28: Lid, sign of d l ~ n , l / d s -
Figure 4.30: Lid, parametric di- rection of constant u.
Figure 4.31: Lid, parametric di- rection of constant v.
CHAPTER 4. ANALYSIS
Figure 4.32: Lid, sign of K,, .
Figure 4.34: Lid, sign d l ~ n , ~ / Q s .
Figure 4.36: Lid, sign of d21~og l~n , l l lds2 .
Figure 4.33: Lid, sign of rcn,.
Figure 4.35: Lid, sign of d ( ~ n , 1
Figure 4.38: Lid, maximum prin- cipal direction.
Figure 4.40: Lid, sign of K ~ , , .
Lid, sign of
Figure 4.39: Lid, minimum prin- cipal direction.
Figure 4.41: Lid, sign of rc,c.
Figure 4.43: Lid, sign of d l ~ m i n I I d s .
SISA? VNV ' P 83LdVH3
CHAPTER 4. ANALYSIS
Figure: 4.46: Pat2, paramer& di- rection of constant a.
Figure 4.48: Pat2, sign of d l b , I p s .
Figure 4.49: Pat2, sign of d2110gI~n, I l / d ~ ~ .
CHAPTER 4. ANALYSIS
Figure 4.50: Bump, parametric direction of constant u.
Figure 4.51 : Bum? %. Figure 4.52: Bump, dl&,, ( I d s .
Figure 4.54: Bump, sign of K,, .
CHAPTER 4. ANALYSIS
Figure 4.55: Bump, sigul gf d l ~ n , 1 Ids.
Figure 4.56: Bump, sign of d2 t h l ~ n , I l Ids2.
Figure 4.57: Improved bump, parametric direction &eonstant a.
Figure 4.58: Improved bump, Enu -
Figure 4.59: Improved bump, dl ~ n , l Ids .
CHAPTER 4. ANALYSTS
Figure 4.60: Improved bump, d2 1109 l ~ n , I 1 I ds2 .
Figure 4.6% a Xmpr~ved bump, sign of K,,. ";
Figure 4.62: Improved bump, sign of I Ids .
Figure 4.63: Improved bump, sign of d2110g1b,Il/ds2.
Figure 4.64: Bump. Figure 4.65: Improved bump.
CHAPTER 4. ANALYSIS 79
Figure 4.66: Saddle 1. Figure 4.67: Saddle 2.
Figure 4.68: Saddle 3. Figure 4.69: Saddle 4.
CHAPTER 4. ANALYSIS
Figure 4.70: Telephone handset 1.
Figure 4.71: Telephone handset 2. Figure 4.72: Telephone handset 3.
CHAPTER 4. ANALYSIS 8 1
Figure 4.73: Mouse top 1.
Figure 4.74: Mouse top 2. Figure 4.75: Mouse top 3.
CHAPTER 4. ANALYSIS
Figure 4.76: Vase 1. Figure 4.77: Vase 2.
Figure 4.78: Vase 3. Figure 4.79: Vase 4.
CHAPTER 4. ANALYSIS
Figure 4.80: Perfume bottle 1. Figure 4.81: Perfume bottle 2.
Figure 4.82: Perfume bottle 3. Figure 4.83: Perfume bottle 4.
Chapter 5
Discussion . %
',
In this chapter. we will discuss how efficient our method is in measuring the aesthetic
quality of parametric surfaces thro~igh analyzing the raw da ta obtained in the previous
chapter.
5.1 The Sample Correlation Coefficient r
From the saddle. telephone handset. mouse top and vase charts (see Figures .5.1 to
3 . 4 ) . we can intuitively tell tha t our method is effective, but not from the perfumc Y
bottle chart . Me would like t o us; a quantitative measure t o show how effective our
met hod is. Thus we resort t o the-sample correlation coefficient.
T h e sample correlation coefficient r is a measure of how strongly related two
variables s and y are in a sample. The sample correlation coefficient for the n pairs
( x l . y, ). .... ( . I - , , . y,) is:
The most important properties of r are as follows [ .5] :
The \ d u e of r does not depend on which of the two variables under s t u d y is
labled s and which is labeled y .
0 The value of r is independent of the units in which s and y are measured.
in the range of [-I , 11.
0 r = 1 if and only if all (a , , y,) pairs lie on a straight line with positive slope, and
r = -1. if and only if all ( x i , y;) pairs lie on a straight line with negative slope.
r measures the degree of linear relationship among variables. A reasonable rule o r
thumb is to say that the correlation is weak if Irl is in the range of [O, 0.51, strong
i f / r / is in the range of [O.S. 11. and moderate otherwise (see [5], p189). We expect il
strong positive linear relationship between the fairness calculated by our method and
the average rank obtained from the subjects. If this relat,ionship exists, it proves that
our method is efficient in measuring the aesthetic quality of parametric surfaces.
In our tests, variable s is the fairness and variable y is the average rank. From the
saddle example, we have 1 pairs of observations (2.47,2.64), (2.09,1.43), (3.66.3.36).
(2.8:3.2..57). According to Equation 5.1 r can be calculated from the five quantities
C x,?C x;. C y,, C y;, C xIyI. these are ll.OTj; 31.87, 10.00, 26.91, 29.08, respecti.vel~..
ThBs the sample correlation coefficient is:
/l'e conclude there is a strong positive linear relationship between the average rank 1 and the fairness in this observed test.
5.2 The ~ o ~ u l a t i o n correlation Coefficient p
The sarnple correlation coefficient r is a measure of how strongly related x and y
are in the observed sample. /Ve can think of the pairs (x,, y,) as having been drawn
from a bivariate population of pairs, with (?(,. 1';) having joint probability distributio~i
j ( x . y ) . The population,correlation coefficient p is a measure of how strongly relatcd
x and y are i n that population. .!n particular, r is a point estimate-for p. and the.
corresponding est irnator is:
CHAPTER 5. DISCliSSIOS A
Is, # - . In the saddle example, the estimate is 0.9072.
The estimated standard deviation of this test" is 0.1803, then for the sample size
n = 4 and the level of significance a = 0.01. t h p a l u e 0.4201 is the boundary of
the upper-tailed rejection region. Since r = 0.9072 is greater than 0.4201. we can
reject the null hypothesis. which is that there is no popiilation correlation between
fairness and average rank at the 0.01 level of significance. Moreover, becausp 0.9072
.is greater than 0.8. we conclude that there is a strong positive linear relationship
between fairness and average rank for all cases.
" From the telephone handset example, we have 3 pairs of observations listed i n
Table 4.13, thus the sahple correlation coefficient is
.I r = 0.8659
The estimated standard deviation of this test is 0.2078. then for the sample'sizc
n = :3 and the level of significance o = 0.01, the value 0.4842 is the boundary of
the upper-tailed rejection region. Since r = 0.8659 is greater than 0.4842, we can f
reject the nul1,hypothesis. which is that there is no population correlation b e t n d n
fairness and average rank at the 0.01 level of significance. hloreover, because 0.5659
is greater than 0.8. we conclude that there is a strong positive linear relatio~~ship
between fairness and average rank for all cases.
~ r & n the mouse top example. we have 3 pairs of observations listed in Table 4.16.
thus the sample correlation coefficient is
The estimated standard de~ia t ion of this test is 0.2161, then for the sample size
n = :3 and the le~.el of significance o = 0.01, the value O.Fj035 is the boundary of
the upper-tailed rejection region. Since r = 0.90.59 is greater than 0..503.5, we car1
j e t the null hypothesis. ~vhich is that there is no population correlation betwrm~
fairness and average rank at the 0.01 level of significance. ,210reover. because O.SO.i!)
is greater than 0.8. we conclude that there is a strong positive linear relatioriship -
between fairness and aiverage rank for all cases.
Frorn the vase example. we have -1 pairs of observatiorls listed in Table 4.19. t h u s
t h e sample correlation coefficient is d
T h e estimated standard deviation of this test is O.lS78, then for t he sample size
n = 1 and the level of significance a = 0.01, the value 0.4376 is the boundary of
t h e upper-tailed rejection region. Since r 4 0.9650 is greater than 0.4376, we can
reject the null hypothesis. which is tha t there is no population correlation between
fairnes.~ and average rank a t the 0.01 level of significance. Moreover, because 0.9650
is greater than 0.8, we conclude tha t there is a strong positive linear relationship '
hetween fairness and average rank for all cases.
Finally. from the perfume bottle example, we have 4 pairs of observations listed
in Table 4.22, thus the sample correlation coefficient is
The estimated-standard deviation of this*,tedt is 0.0.571, then for the sample size ' I
n = 1 and the level of significance a = 0.01, t k value 0.1330 is t he boundary of the
upper-tailed rejection region. Since r = 0.0100 is less than 0.1330, we cannot reject
the null hypoth-esis, which is tha t there is no population correlation between fairness
and average rank a t the 0.01 level of significance..
5.3 More Analysis
S o LR re have 5 sample correlation coefficients ( x l , ... x,,) = (0.9072, 0.56.59, 0.90.59.
O.96.50. 0.0100). If we use an estimator - Y / T ~ . then we can get an average sarnplv
correlation coefficient.
?'hi5 shows the positive linear relationship between the fairness and the average rank
moderate and close to strong.
The main reason that the perfume bottle example fails is tha t the difference in
shape between perfume bottle number 4 and the other three is much more obvious
than the difference in shape between perfume bottle number 1. 2 and 3. This difference
is not measurable using our method because it is like the difference in shape between
an apple and an orange rather than the difference in shape between a nice looking
apple and a bad looking apple. Our method is designed to measure similar shapes.
Thus before using our method, i t might be helpful to clasify the s ~ r f a c e s ~ b y clustering
similar ones according to some similarity measure, and we would suggest this be
considered as future work. A similarity measure for 3D shape models based on the
correspondence of the points on the models is proposed by Kawabata [18], and more
discussion about this topic could be found in [23].
If we take out perfume bottle number 4 from our test, the sample correlation
coefficient for perfume bottle example will be .
The estimated standard deviation of this test is 0.1998, then for the sample size r l = ;3
and the level of significance (I = 0.01, the value 0.4655 is the boundary of the upper-
tailed rejection region. Since r = 0.8266 is greater than 0.46.55, we can reject the
null hypothesis, which is that there is no population correlation between fairness and
average rank at the 0.0 1 level of significance. Moreover, because 0.8266 is greater t hall
0.8, we conclude that there is a strong positive linear relationship between f a i r ~ i e s h
and average rank for all cases.
Me u-ill also have a new average sample correlation coefficient.
C
The sample correlation coefficients (with and without perfume bottle # 4 ) are listed
i n Table 5.1. This table shows the positive linear relationship between the fairness
and the average rank is strong. thus our method is effective i n measuring the aesthetic
quality of parametric surfaces with similar shapes, which is our final conclusion.
CHAPTER 5. DISCUSSION
Saddle Telephone handset Mouse top
Tnew
Without T
With perfume bottle # 4
0.9072 0.8659 0.9059
Vase Perfume bottle Average
Table 5.1: Different average sample correlation coefficients with and without perfume bottle # 4.
0.9650 0.0100 0.7308
EI Fairness
1 Standard error
Figure 5.1: Fairness and average rank for saddles.
CHAPTER 5. DISCUSSION
E! Fairness
Average rank
] Standard error
Figure 5.2: Fairness and average rank for telephone handsets.
b!i~ Fairness
1 Standard error
Figure 5.3: Fairness and average rank for mouse tops.
CHAPTER 5. DISCUSSION
- #4 #2 #I #3
Figure 5.4: Fairness and average ra
Fairness
] Standard error
,nk for vases.
I rn Average rank I
] Standard error
Figure 5.5: Fairness and average rank for perfume bottles.
Chapter 6
Conclusion
6.1 Summary
In this thesis, we first reviewed some surface analysis and surface synthesis methods in
Chapter I-. Then in Chapter 2 we introduced related differential geometry knowledge.
on curves and surfaces. In Chapter 3.3.1, the formulae were derived t o compute the.
first and second derivatives of norrhal curvature with respect t o arc length in all!
.b
arbitrary direction, while in C'hapter 3.3.2 we derived the formulae t o compute the
first and second derivatives of Gaussian curvature with respect t o arc length in an!
arbitrary direction. A11 these formulae can be applied t o parametric surfaces such as
Bezier. B-spline and NI'RBS surfaces. In Chapter 4, we gave a heuristic formula t o
compute the fairness of networks of isoparametric curves.
111 our implementation. we can compute the first and second derivatives of nor-
mal and Gaussian curvatures for three kinds of characteristic curves: planar curves.
isoparametric curves and lines of curvature of cubic Bezier and B-spline surfaces. CCr.
can also compute a measure of the fairness of networks of isoparametric curves of c ~ t -
bic Bezier and B-spline surfaces. Several methods have been implemented to visualize
geometric properties of parametric surfaces and to detect potential design problems.
One exponential color map method has been implemented to show the values of ~111.-
vat ltres and their derivatives. The average number of segments of constant s ip) of
Several experiments we have carried out were described in Chapter 4 and the
analysis of these experiments was discussed in Chapter 4 and 5. =a
6 . 2 Conclusion
The goal of our work was first to develop a method t o inspect geometric properties
of pxamet r i c surfaces and detect potential design problems, and second, and more
importantly, t o develop a quantitative measure of the aesthetic quality of parametric
surfaces. From the discussion in Chapters 4 and 5,)i t is'shown tha t using our method.
we can inspect geometric properties of parametric surfaces using color t o map normal
curvature, Gaussian curvature and the first and second derivatives of normal and
Gaussian curvatures. We can also detect potential design problems through examining
the change of the sign of normal curvature, Gaussian curvature and their first and
second derivatives. l i e have also developed a quantitative method for evaluating
aesthct ic qualities in shape using derivatives of normal and Gaussian curvatures. based
on the computation of normal curvature, Gaussian curvature and their first and s e c o ~ ~ t l
deriratives, we compared it t o choices made by human subjects, and found a high
degree correlation. /
\\'e h a w developed a quanti tative method for evaluating aesthetic clualities i l l
shape using derivatives of normal anti Gaussian curvatures.
6.3 Future Work
\\-e would like t o see the implementation of computing the fairness of N U K B S surfaces.
This will let us a n d y z e some surfaces in the industrial design world.
Isoparametric curves are dependent on the paramet,rization of surfaces. 'l'hus
thc fairness of netivorks of isoparanietric curves is not unique when using diffcw111
parametrizations t o represent t h e same surface. Lines of curvature are riot depe~ltler~t 1
0 1 1 the parametrization of surfaces. One interesting direction is to explore the fairness
of networks of lines of curi.ature. which could be a better measurement of the aesthetic
quali t of parametric surfaces.
I
Our method measures the aesthetic quality of parametric surfaces; it would be . interesting and valuable to see if our method could be recast as a set of constraints
w
for automating the design of fair surfaces. '
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List of Symbols
b ,qnit curve binormal vector, 17
\
f t h e f rame f& a space curve. I T
F surface curve f rame. 20
n .principal normal vector for curve. 16
N surface norinal vector. 22
r position vector for a curve or surface. 16. '-12
d
t tangent 1,ector of a cur1.e. 16
t i , niean (al 'erage) curvature. 2.5 4
t ic; Gaussian cur1,ature. 2.5. 3.5
tic;* first derivative of Gaussian curi7ature. 3.5
t ic ; . . second derivative of Caussian curvature. 36
ti,,,, rnax i rnu~n principal cur1,ature. 26
t i n i , , minimum principal cu r ix tu re . 26
K,, normal curvature, 21. 24
s a K , fir& derivative of normal c u r v a h r e , 31
Q
I 3
K~~ second derivative of normal curvature, 3%
T torsion of a curve. 19