, 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