TRIGONOMETRIC BEZIER TRANSITION CURVES AND APPLICATIONS

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TRIGONOMETRIC BEZIER TRANSITION CURVES AND APPLICATIONS

MD YUSHALIFY MISRO

UNIVERSITI SAINS MALAYSIA

2017

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TRIGONOMETRIC BEZIER TRANSITION CURVES AND APPLICATIONS

by

MD YUSHALIFY MISRO

Thesis submitted in fulﬁlment of the requirements for the degree of

Doctor of Philosphy

August 2017

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ACKNOWLEDGEMENT

In the name of Allah, the Most Gracious and Most Merciful. All praises to Allah and peace be upon His beloved prophet MUHAMMAD s.a.w. I am grateful for all the strength and blessings He bestowed me in order for me to complete my research. I was blessed to be backed up by wonderful people throughout this entire research. I would like to express my gratitude to all of them. My deepest gratitude goes to my supervisor, Dr. Ahmad Lutﬁ Amri Ramli for his guidance, moral injections, and encouragement at every stage of this research, especially during my falls. It has always been an honor to work with him. I am thankful for his commitments and contributions of ideas. I also appreciate the freedom he gave for me to make my own decision regarding this research. My sincere admiration goes to Prof. Jamaludin Md Ali, senior professor in Universiti Sains Malaysia (USM) for his endless encouragement since my degree years until now. His enthusiasm in this research area motivates me to follow his path.

My special thanks goes to Prof Hailiza Kamarulhaili for allowing me to pursue my doctoral studies and providing me a chance to work in this healthy research environ- ment. I would also like to acknowledge the Ministry of Higher Education Malaysia for conferring me a sufﬁcient scholarship throughout this research period. No words can I express how indebted I am to my parents, Hamtiah Md Suadi and my late father, Misro Kasiran who were always there to support me even while battling stage 4 of cancer.

Their endless love, prayers, and priceless time have made me who I am today. Also, thanks to all my siblings and friends who stayed during the thick and thin. Our memo- ries will always be treasured. Lastly, thanks to the non academic staff of the school all for their help and cooperation.

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LIST OF TABLES

Page Table 3.1 Curvature value comparison for blue and red curve 36

Table 3.2 Curvature value for second curve 36

Table 6.1 Speed Approximation Based on Manual Attempt 128 Table 6.2 Speed approximation on 10 control points of Tun Sardon

Road

131

Table 6.3 Curvature values, Velocity Designs, Color Gradients for 1^st curve of planar quintic trigonometric Bézier

134

Table 6.4 Curvature values, Velocity Designs, Color Gradients for 2^nd curve of planar quintic trigonometric Bézier

135

Table 6.5 Curvature values, Velocity Designs, Color Gradients for 3^rd curve of planar quintic trigonometric Bézier

136

Table 6.6 Curvature values, Velocity Designs, Color Gradients for 4^th curve of planar quintic trigonometric Bézier

137

138

139

140

141

142

Table 6.12 Comparison of speed estimation of quintic Bézier curves and quintic trigonometric Bézier curves

146

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LIST OF FIGURES

Page Figure 3.1 Cubic trigonometric Bézier basis function with different val-

ues of shape parameters

22

Figure 3.2 Quintic trigonometric Bézier basis function with different values of shape parameters

27

Figure 3.3 A curve withp=q(left) and its curvature distribution (right) 29 Figure 3.4 A curve with ﬁxed p(left) and its curvature distribution

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29

Figure 3.5 A curve with ﬁxedq(left) and its curvature distribution (right)

29

Figure 3.6 Composite quintic trigonometric Bézier curve 35 Figure 3.7 Clover design using quintic trigonometric Bézier composite

curve

38

Figure 3.7(a) Red clover 38

Figure 3.7(b) Blue clover 38

Figure 3.7(c) Brown clover 38

Figure 3.7(d) Purple clover 38

Figure 3.7(e) Green clover 38

Figure 4.1 J-shaped transition curve 44

Figure 4.2 Curvature proﬁle of J-shaped transition curve (left) and its curvature derivative (right)

44

Figure 4.3 J-shaped trigonometric spiral curve 49

Figure 4.4 J-shaped trigonometric spiral curves with shape parameters p=1,q=0 (red-dashed curve) and p=−1,q=0 (blue- dotted curve)

49

Figure 4.5 Curvature proﬁle with shape parameters p=1,q=0 (red- dashed curve) and p=−1,q=0 (blue-dotted curve)

50

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Figure 4.6 J-shaped trigonometric spiral curve with different values of

θ 50

Figure 4.7 Curvature proﬁle of J-shaped spiral with differentθ values 51 Figure 4.8 Straight line to straight line spiral curve 53 Figure 4.9 Straight line to straight line spiral trigonometric Bézier curve

with the shape parameters p=−2,q=0 (red-dashed) and p=1,q=0 (blue-dotted)

53

Figure 4.10 S-shaped curve composed by two cubic trigonometric Bézier spiral

55

Figure 4.11 An S-shaped transition curve with same radii (left) and its curvature proﬁle (right)

58

Figure 4.12 S-shaped transition curves of same radii with shape parametersp=1,q=1 for black-lined,p=0,q=0 for red-dashed and p=−1,q=−1 for blue-dotted (left) and its curvature proﬁle (right)

59

Figure 4.13 Curvature derivatives of S-shaped transition curves of same radii

60

Figure 4.14 S-shaped transition curves of same radii with shape parameters p=0,q=1 for red-lined and p=1,q=0 for blue- dotted (left) and its curvature proﬁle (right)

61

Figure 4.15 An S-shaped transition curve of different radii (left) and its curvature proﬁle (right)

61

Figure 4.16 S-shaped transition curves in different radii with shape parametersp=−1,q=−1 for black-lined curve,p=0,q=0 for red-dashed curve and p=1,q=1 for blue-dotted curve (left) and their curvature proﬁles (right)

62

Figure 4.17 Curvature derivatives of S-shaped transition curves of different radii

63

Figure 4.18 S-shaped transition curves with pairs of shape parameters p=1,q=0 for red curve and p=0,q=1 for blue-dotted curve (left) and their curvature proﬁles (right)

64

Figure 4.19 C-shaped curve composed by two cubic trigonometric Bézier spirals

65

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Figure 4.20 A C-shaped transition curve of same radii (left) and its curvature proﬁle (right)

67

Figure 4.21 C-shaped transition curves with shape parametersp=1,q= 1 for blue-dotted curve, p=0,q=0 for red-dashed curve andp=−1,q=−1 for black line curve (left) and their curvature proﬁles (right)

69

Figure 4.22 Curvature derivatives of C-shaped transition curves of same radii

70

Figure 4.23 C-shaped transition curves with shape parametersp=1,q= 0 for blue-dotted curve and p=0,q=1 for red-lined curve (left) and its curvature proﬁle (right)

71

Figure 4.24 A C-shaped transition curve of different radii (left) and its curvature proﬁle (right)

71

Figure 4.25 C-shaped transition curves with shape parametersp=1,q= 1 for blue-dotted, p = 0,q = 0 for red-dashed, and p =

−1,q=−1 for black-lined curve (left) and their curvature proﬁles (right)

72

Figure 4.26 Curvature derivatives of S-shaped transition curves of different radii

73

Figure 4.27 C-shaped transition curves with shape parametersp=0,q= 1 for red-dotted andp=−1,q=1 for blue-dashed (left) and its curvature proﬁle (right)

73

Figure 4.28 A C-shaped trigonometric Bézier spiral curve (left) and its curvature proﬁle (right)

74

Figure 4.29 A C-shaped trigonometric Bézier transition curve (left) and its curvature proﬁle (right)

75

Figure 4.30 A C-shaped trigonometric Bézier transition curve with same pairs of shape parameters (left) and their curvature proﬁles (right)

76

Figure 5.1 A J-shaped spiral quintic trigonometric Bézier curve 84 Figure 5.2 Curvature proﬁle of J-shaped spiral quintic trigonometric

Bézier curve (left) and its curvature derivative (right)

85

Figure 5.3 V-shaped composed of two PH quintic trigonometric Bézier spiral curves

87

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Figure 5.4 V-shaped composed of two spiral curves without control points

87

Figure 5.5 S-shaped spiral curve with(m₀,m₁) = (3.293,1.293) 89 Figure 5.6 S-shaped spiral curve withS_red= (m₀,m₁) = (2.707,1.707),

S_blue = (m₀,m₁) = (1.8,1) and S_black = (m₀,m₁) = (3.293,1.293)

90

Figure 5.7 Quintic trigonometric Bézier S-shaped transition curve on same radii with shape parameters p=1,q=1

92

Figure 5.8 Curvature proﬁle of S-shaped transition curve on same radii (left) and its curvature derivative (right)

92

Figure 5.9 Quintic trigonometric Bézier S-shaped transition curves on two different radii of circles

93

Figure 5.10 Curvature proﬁle of quintic trigonometric Bézier S-shaped transition curve on two different radii (left) and its curvature derivative (right)

93

Figure 5.11 Quintic trigonometric S-shaped transition curves using different parameterm

94

Figure 5.12 Curvature proﬁle of quintic trigonometric S-shaped transition curves using different parameterm(left) and their curvature derivatives (right)

95

Figure 5.13 Quintic trigonometric Bézier S-shaped transition curves with u₀=2.5 (blue-dashed curve),u₀=2.0 (black-dotted curve) andu₀=1.5 (red-lined curve)

95

Figure 5.14 Curvature proﬁle of quintic trigonometric S-shaped transition curves using different parameteru₀(left) and their curvature derivatives (right)

96

Figure 5.15 Quintic trigonometric Bézier S-shaped transition curves with m=2.2 (blue-dashed curve), m=1.7 (black-dotted curve) andm=1.2 (red-lined curve)

97

Figure 5.16 Curvature proﬁle of PH quintic trigonometric S-shaped transition curves using different parameterm(left) and their curvature derivative (right)

97

Figure 5.17 PH Quintic trigonometric S-shaped transition curve using different parameteru₀

98

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Figure 5.18 Curvature proﬁle of PH quintic trigonometric S-shaped transition curve using different parameteru₀(left) and its curvature derivative (right)

99

Figure 5.19 C-shaped spiral curve withC= (m₀,m₁) = (2.707,1.707) 100 Figure 5.20 C-shaped spiral curve withC_red= (m₀,m₁) = (2.707,1.707),

C_blue = (m₀,m₁) = (1.8,1) and C_black = (m₀,m₁) = (3.293,1.293)

101

Figure 5.21 C-shaped transition curve on same radii 103

Figure 5.22 Curvature proﬁle of C-shaped transition curve on same radii (left) and its curvature derivative (right)

104

Figure 5.23 Quintic trigonometric Bézier C-shaped transition curve with shape parameters p=1,q=1

104

Figure 5.24 Curvature proﬁles of quintic trigonometric Bézier C-shaped transition curve and its curvature derivative (right)

105

Figure 5.25 Quintic trigonometric Bézier C-shaped transition curve with m=1.8 (blue-dashed curve), m=1.5 (black-dotted curve) andm=1.2 (red-lined curve)

106

Figure 5.26 Curvature proﬁles of quintic trigonometric C-shaped transition curves using different parameterm(left) and their curvature derivatives (right)

107

Figure 5.27 Quintic trigonometric C-shaped transition curve using dif-

ferent values ofθ 107

Figure 5.28 Curvature proﬁles of quintic trigonometric C-shaped transition curves using different parameterθ (left) and their curvature derivative (right)

108

Figure 5.29 Quintic trigonometric C-shaped transition curves using different values ofm

108

Figure 5.30 Curvature proﬁles of quintic trigonometric C-shaped transition curves using different parametersm(left) and their curvature derivatives (right)

109

Figure 5.31 Quintic trigonometric Bézier C-shaped transition curve with θ =55 degree (blue-dashed curve), θ =50 degree (black- dotted curve) andθ =45 degree (red-lined curve)

109

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Figure 5.32 Curvature proﬁle of quintic trigonometric C-shaped transition curves using different parameterθ (left) and their curvature derivatives (right)

110

Figure 5.33 Circle in circle spiral curve 111

Figure 5.34 Curvature proﬁle of circle in circle spiral curve (left) and its curvature derivative (right)

112

Figure 5.35 Circle in circle transition curve 113

Figure 5.36 Curvature proﬁle of circle in circle quintic trigonometric Bézier transition curve (left) and its curvature derivative (right)

114

Figure 6.1 A wholly transition curve 118

Figure 6.2 A compound transition curve 118

Figure 6.3 Two types of clothoids (Retrieved from Song (2006)) 119 Figure 6.4 A reverse curve (Retrieved from Song (2006)) 120 Figure 6.5 A circle in circle curve (Retrieved from Song (2006)) 120

Figure 6.6 A transitional section of curve 122

Figure 6.7 Superevelation in highway design 124

Figure 6.8 Topography Tun Sardon Road (Retrieved from Veloviewer) 124 Figure 6.9 3D Tun Sardon Road Elevation (Retrieved from Veloviewer) 125 Figure 6.10 Tun Sardon Road Elevation (Retrieved from Veloviewer) 126

Figure 6.11 Research Area 127

Figure 6.12 Quintic Bézier Curve Mapping on Road 132

Figure 6.13 1st piecewise of quintic trigonometric Bézier curve 134 Figure 6.14 2nd piecewise of quintic trigonometric Bézier curve 135 Figure 6.15 3rd piecewise of quintic trigonometric Bézier curve 136 Figure 6.16 4th piecewise of quintic trigonometric Bézier curve 137 Figure 6.17 5th piecewise of quintic trigonometric Bézier curve 138

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Figure 6.18 6th piecewise of quintic trigonometric Bézier curve 139 Figure 6.19 7th piecewise of quintic trigonometric Bézier curve 140 Figure 6.20 8th piecewise of quintic trigonometric Bézier curve 141 Figure 6.21 9th piecewise of quintic trigonometric Bézier curve 142 Figure 6.22 Interpolation of quintic trigonometric Bézier curve with

color gradient schemes corresponding to maximum speed estimation

143

Figure 6.23 Interpolation of quintic trigonometric Bézier curve for all curve segments with shape parameters p=−4 andq=1 for the ﬁrst curve segment and alternately for consecutive curves

144

Figure 6.24 Interpolation of quintic trigonometric Bézier curve for all curve segments with shape parameters isp=1 andq=−4 for the ﬁrst curve segment and alternately for consecutive curves

145

Figure 1 Appendix A1 162

Figure 2 Appendix A2 163

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LIST OF ABBREVIATIONS

CAGD Computer Aided Geometric Design CAD Computer Aided Design

CAM Computer Aided Manufacturing USM Universiti Sains Malaysia PH Pythagorean Hodograph JKR Jabatan Kerja Raya

CNC Computer Numerical Control

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LIST OF SYMBOLS

× cross product

· dot product

√ square root

a vector a parallel

θ angle in degree μ angle in degree α angle in degree β angle in degree

Pi control points of P Q_i control points of Q B_i control points of B p shape parameter left q shape parameter right s free parameter

m free parameter n free parameter r radius of circle t variable

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t₀ knot point t₁ knot point

T₀ beginning unit tangent T₁ ending unit tangent

C⁰ position parametric continuity C¹ tangent parametric continuity C² curvature parametric continuity G⁰ position geometric continuity G¹ tangent geometric continuity G² curvature geometric continuity

f_i cubic trigonometric Bezier basis function

g_i quintic trigonometric Bezier basis function ω0 ﬁrst circle

ω1 second circle

∞ inﬁnity

∑ summation λ lambda π pi

κ(t) curvature

κ(t) curvature derivative

υ(t) numerator of curvature derivative

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u(t) quadratic function with coefﬁcientu_i v(t) quadratic function with coefﬁcientv_i z(t) parametric curve

h=P₀P₁ magnitude fromP₁toP₀ k=P₃P₂ magnitude fromP₃toP₂ C_m cosine basis function

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LENGKUNG PERALIHAN BEZIER TRIGONOMETRI DAN APLIKASI

ABSTRAK

Tesis ini menghuraikan pembinaan lengkung Bézier kuintik trigonometri dengan dua parameter bentuk melalui perbincangan fungsi asas, sifat, dan komposisi dua se- gmen lengkung. Lima templat lengkung peralihan yang boleh lentur dijana menggunakan lengkung trigonometri kubik dan kuintik dengan kehadiran parameter bentuk.

Lima templat lengkung peralihan diperoleh dengan menghubungkan garis lurus kepada bulatan, bulatan kepada bulatan dengan peralihan C, bulatan kepada bulatan dengan peralihan S, garis lurus kepada garis lurus, dan bulatan kepada bulatan yang mem- punyai bulatan kecil di dalam bulatan besar. Dua daripada templat tersebut membincangkan lengkung peralihan berbentuk S dan C menggunakan satu lengkung peralihan atau yang terbentuk daripada dua lingkaran Bézier. Tesis ini turut membincangkan aplikasi dalam reka bentuk lebuh raya dengan membuat anggaran halaju jalan raya menggunakan maklumat kelengkungan. Perbandingan antara lengkung Bézier kuintik dan lengkung Bézier kuintik trigonometri terhadap set data sebenar turut dibincangkan.

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TRIGONOMETRIC BEZIER TRANSITION CURVES AND APPLICATIONS

ABSTRACT

This thesis describes the development of a new quintic trigonometric Bézier curve with two shape parameters by discussing its basis function, properties, and composition of two curve segments in the early part of this thesis. Five ﬂexible templates of transition curves are generated using both cubic and a newly proposed quintic trigonometric Bézier curves with the presence of shape parameters. The ﬁve templates of transition curves are achieved by joining a straight line to a circle, circle to circle with a broken back C transition, circle to circle with an S transition, straight line to straight line, and circle to circle where small circle lies inside the big circle. Two of the templates will discuss on S-shaped and C-shaped transition curve using single transition curve and two Bézier spiral segments. This thesis also includes works on the application in highway design by approximating maximum speed estimation using curvature information.

Comparisons between quintic Bézier curve and quintic trigonometric Bézier curve on real data sets are also presented.

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CHAPTER 1 INTRODUCTION

1.1 Introduction

Computers have become an indispensable tool in modelling and simulation. So- phisticated geometric shapes and physical objects are modelled in the domain of complex physical environments. As computational power drastically increased, users and applications also demand for a simpler method with less computational time. Ap- proaches for object modelling and designing involves non-physical methods. Non- physical method can be deﬁned as individual or groups of control point are manually adjusted for shape editing and designing (Gibson and Mirtich, 1997).

Computer Aided Geometric Design (CAGD) is a discipline dealing with computational geometry. Some applications of CAGD are construction and manipulation of free-form curves and surfaces (Farin, 2002a). The major breakthrough in CAGD were the theory of Bézier curves and surfaces, as well as B-spline method. B-spline actually originated from the word ‘spline’. Spline was not actually a mathematical term but was previously a draftsmen’s tool that consists of a ﬂexible strip of metal or wood and several heavy pieces which hold or anchored the ﬂexible strips for drawing ship (Farin, 2002b).

In 1959, De Casteljau had already developed curves similar to Bézier indepen- dently, but his work were never published. Therefore, when Pierre Étienne Bézier from a French car manufacturer, Renault, presented a parametric polynomial curve

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representation technique in 1962, the whole theory of polynomial curves and surfaces in Bernstein form is now using Bézier’s name (Farin, 2002b). Bézier and B-spline are two important designing techniques with roots in the late 1950’s. Both of these techniques are used for interactive curve and surface design (Faux and Pratt, 1979;

Hoschek et al., 1993; Mathews, 1986).

Bézier and B-spline are mostly used for design purposes that will be incorporated and integrated into Computer Aided Design (CAD) or Computer Aided Manufacturing (CAM). Bézier is mostly useful in CAD/CAM to construct and manipulate free-form curves and surfaces to fabricate products such as automotive, aerospace and shipbuild- ing. Desired shape features play a pivotal role in constructing product of curves and surfaces. In some cases, it is difﬁcult to distinguish between Bézier and B-spline because any Bézier curve of arbitrary degree can be converted into B-spline and any B-spline can be transformed into one or more Bézier curves(Prautzsch et al., 2013;

Romani and Sabin, 2004).

In terms of providing a local control on the shape of the curve, B-spline is a better choice compared to Bézier, where Bézier offers a global shape control. B-spline curve is more ﬂexible and a better piecewise than Bézier curves. Therefore, B-spline is a powerful generalization of Bézier curves (Racine, 2014). Although, with all the advantages of B-spline curves, it does have a shortcoming. B-spline are more complex than Bézier curve (Parekh et al., 2006) and thus industrial designers would prefer to use Bézier in terms of practicality. Several problems in applied mathematics and engineering would require splines to be one of their solutions.

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This includes CAGD (Hoschek et al., 1993), function approximation (Nuernberger, 1989), optimal control (Schumaker, 2007), data ﬁtting (Dierckx, 1995) and many more. Transition curve is useful in various ﬁelds, especially CAGD. It is also an important part in generating round corners or smooth transition between two circles.

Properties of transition curves which have gradual increasing or decreasing curvature between circles plays an important role in designing highways and railways where a transition path is needed to lead into one part of the road to another such as from one straight line to circle, circle to circle, and circle to straight line.

In road and rail design, passenger’s comfort and safety is another important aspect to be considered. Transition curve may ease up the transition, prevent accidents caused by sharp changes in direction, and reduces shaking of vehicles in operation.

It also contributes in smoother, stable and safer transition. Baass (1984) proposed a design technique by implementing single ordinary Bézier function of continuity with ﬁve template of transition curve.

• Type 1. Straight line to circle with a J-shaped transition curve (Habib and Sakai, 2005b; Walton et al., 2003)

• Type 2. Circle to circle with a broken back C-shaped transition (Habib and Sakai, 2007a; Yahaya et al., 2008)

• Type 3. Circle to circle with an S-shaped transition (Habib and Sakai, 2007a;

Walton and Meek, 1999; Yahaya et al., 2009)

• Type 4. Straight line to straight line (Habib and Sakai, 2005b; Walton et al., 2003)

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• Type 5. Circle in circle with C-shaped transition (Walton and Meek, 1996; Ya- haya et al., 2010)

Previous works that discussed on the construction of S-shaped transition curve had been done using cubic Bézier (Habib and Sakai, 2003; Walton and Meek, 1999) and cubic Bézier-like (Yahaya et al., 2008) and cubic trigonometric Bézier (Abbas et al., 2011). Walton and Meek (1999) constructed C-shaped transition curve using cubic Bézier curve whereas Habib and Sakai (2003) constructed transition curve but using a different approach. Other authors who had constructed C-shaped transition curve used a cubic Bézier-like curve such as in Yahaya et al. (2008). They used three control points which produced two segments.

Throughout this thesis, transition curves will be researched extensively using cubic trigonometric Bézier curves and quintic trigonometric Bézier curves on all ﬁve templates of transition curves. Cubic Bézier curve is the lowest degree of polynomial which can be used as transition curve to construct or design in CAD/CAM, whereas quintic curve is the lowest degree of Pythagorean Hodograph curve that may have an inﬂection point, as required for S-shaped transition curve (Habib and Sakai, 2003).

Single cubic curve can be used as transition curve between two circles with the guarantee that S-shaped transition curve does not have interior curvature extremum and C-shaped transition curve does not have inﬂection point or singularities. Even though cubic curve is smoother and simpler, it is not always helpful since they may have unwanted inﬂection points and singularities (Habib and Sakai, 2007b). A cubic segment has the following undesirable features which are; its arc-length is the integral

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part of the square root of polynomial of its parameter, its offset is neither polynomial nor a rational algebraic function of its parameter, and it may have more curvature extrema than necessary.

Pythagorean Hodograph curve does not suffer from the ﬁrst two undesirable features that were mentioned before. Pythagorean originally came from the word Pythago- ras, where is often referred to a pure mathematician (Ratner, 2009). Pythagoras contributed to studies of areas, polygons, proportions, and also perfect numbers. Pythagorean Hodograph curves were developed by Farouki (2008). Hodograph is deﬁned as derivatives ofz(t)is the square of a complex polynomial. PH quintic curve is the isomorphic shape to general cubic curve.

1.2 Motivation

Bézier technique is a powerful tool that has been frequently used to generate free form curves and surfaces. It is also used for approximating, interpolating, preserving shape, designing shape of curves. Trigonometric Bézier acts as an alternative useful tools rather than tedious conventional rational Bézier which both are used as mediums to control desired curves. Trigonometric offers more ﬂexibility by executing the results in simpler ways than rational Bézier.

Trigonometric Bézier curve is a useful function that inherits nice properties similar to rational Bézier, except the shape parameters are better in terms of less complexity, less time consuming, and less tedious works compared to rational Bézier. Transition curves are being constructed using Bézier, Bézier-like, Log aesthetic curves and also trigonometric Bézier. In this thesis, further discussion will be done on ﬁve templates

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of transition curve using cubic and quintic trigonometric Bézier curves. The reason for the use of cubic and quintic trigonometric Bézier curves is due to trigonometric Bézier curves itself needs to satisfy some geometrical properties such as partition of unity and symmetry, therefore in order to have a more flexible curve, we need to have two shape parameters in its basis function. So when quartic trigonometric Bézier curve is used, it can only have five control points and its very difficult to control the curve because the properties of Bézier curves require it to be in symmetrical form in order to satisfy the conditions of transition curves. Some numerical analysis will also be provided by applying the function on real data sets.

1.3 Objectives

This thesis deals with the development of a new function called quintic trigonometric Bézier curve with two shape parameters and also the construction of ﬁve templates of transition curve by using cubic and quintic trigonometric Bézier curve as transition curves. The objectives of this research are:

1. To propose a new generalized quintic trigonometric Bézier curve with two shape parameters by discussing its basis function, properties and composition of two curve segments.

2. To generate ﬁve ﬂexible templates of transition curves using cubic trigonometric Bézier curve and a newly proposed quintic trigonometric Bézier curve with the presence of two shape parameters.

3. Applications in highway design by approximating maximum speed estimation

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1.4 Outline of Thesis

The description of the chapters included in this thesis are as follows:

Chapter 2 presents an overview of parametric curves including Bézier and trigonometric Bézier. In this section, parametric and geometric continuity are discussed with the important use of curvatures. Some notations and terminology that are essential to understand the work, detailed literature reviews and related works are presented.

In Chapter 3, the cubic trigonometric Bézier function and its properties are discussed. A new function of quintic trigonometric Bézier function with its properties respectively will also be proposed and demonstrated in this chapter.

Chapter 4 deals with the construction of ﬁve templates of transition curve using cubic trigonometric Bézier curve. We will also discuss further on two templates which are S-shaped and C-shaped that can be represented by two piece curves or single curve, where one is classiﬁed as spiral while another one as transition curve.

In Chapter 5, the construction of ﬁve templates of design techniques using quintic trigonometric Bézier curves will be presented. Quintic trigonometric Bézier curves are isomorphic shape with cubic trigonometric Bézier curve that possessed almost similar features to cubic curve, but with more attractive properties.

Application on real data set using quintic Bézier and quintic trigonometric Bézier along their comparisons will be demonstrated in Chapter 6. Some classical theory with some different approaches to calculate maximum speed estimation are discussed.

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Finally, the ﬁnal conclusion followed by some suggestions for future researches will be provided in Chapter 7.

1.5 Problem Statements

1. To obtain a ﬂexible curve by using trigonometric Bézier without having to use weightage in its denominator in order to alter the curve.

2. Previous author (Abbas et al., 2010), use cubic trigonometric Bézier curve to construct only one template of transition curve which is Type III without further discussion on different pairs of shape parameter or different radii of circles.

Therefore, this research attempt to complete the gaps left by previous author.

3. The quintic trigonometric Bézier curve have never been applied on Pythagorean Hodograph function using the same approaches and the use of this function in constructing ﬁve templates of transition curve were never discussed in any previous researches.

1.6 Scope and Limitations

1. This research is focused on two-dimensional space as a basic guideline to form a strong foundation before it can be extended into a three-dimensional space of transition curve using both cubic and quintic trigonometric Bézier curves.

2. For the purpose of this research and to answer research objectives, the application of this transition curve is focused only on highway designing and approximating maximum speed estimation.

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CHAPTER 2 BACKGROUND AND LITERATURE REVIEW

2.1 Parametric Curves

Based on the book A History of Curves and Surfaces by Farin (2002b), curves were originally constructed by draftsmen. Most of the time, they use circles as their basic tool. French curve is a set of templates that function as common tools when they had to draw some of the ’free forms’. Meanwhile, wooden curves were designed carefully and consist of pieces of conics and spirals. Other mechanical tools such as spline was also used. This spline is a ﬂexible strips of wood that was held in place and shape by metal weights known as ducks. The mechanical counterpart to a mechanical spline is a spline curve, one of the most fundamental parametric curve forms.

Later in the late 1800s, Serret/Frenet documented about the differential geometry of parametric curves. J. Ferguson and D. MacLaren constructed curves for the design of wings for US aircraft company Boeing in the middle of 1950s. They formed combined curves by piecing cubic space curves together which were overall twice differentiable.

Ferguson also used the cubic Hermite form which deﬁnes a cubic in terms of two endpoint and two endpoint derivatives. The most fundamental parametric curve forms are the Bézier curves (Farin, 2002b).

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2.1.1 Bézier curve

Pierre Étienne Bézier’s initial idea was to represent a basic curve as the intersection of two elliptical cylinders. Later, Bézier moved to polynomial formulations of this inital concept and also extended it to higher degrees. Bézier’s work were widely published and caught A.R Forrest’s attention. His works on Bézier curves helped to popularize and make Bézier curves inﬂuential. He then realized that Bézier curves can be expressed in terms of Bernstein polynomial. Several Bernstein basis properties control the behaviour of Bézier curves like symmetry, recursion, non-negativity, partition of unity, variation diminishing property, derivatives and so (Farin, 2002b).

Han et al. (2008) have achieved similar results by extending the classical Bernstein basis function that inherits most properties of Bézier curve. The basis function was introduced with n adjustable shape parameters λ to control the shape of the curve without changing the control polygon. From the view point of basis function, Bézier curves were also investigated using trigonometric polynomials.

2.1.2 Trigonometric Bézier curve

Bézier curves had been investigated and expanded using trigonometric polynomials (Fitter et al., 2014). Parametric representation through trigonometric polynomial had been emphasized for its beneﬁt in offering local control over the shape through shape parameters and its continuity conditions are being examined consistently under various degrees like quadratic trigonometric Bézier curve (Bashir et al., 2012), cubic trigonometric Bézier curve (Han et al., 2009), quartic trigonometric Bézier curve (Dube and Sharma, 2013), quasi-quintic trigonometric Bézier curves (Bashir et al., 2013).

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A recent development in CAGD is the introduction of trigonometric spline based on trigonometric polynomials and a blending of algebraic and trigonometric polynomials. Schoenberg (1964) was the pioneer in presenting trigonometric spline whose components were piecewise trigonometric polynomials. This introduction of trigonometric splines is a new beginning for researchers. Expanding previous works, Lyche and Winther (1979) established a three term recurrence relation for trigonometric B- splines of arbitrary order, analogous to that for polynomial B-splines. The method to differentiate trigonometric B-splines and a trigonometric analog of Marsden’s identity were also given (Hoschek et al., 1993).

The trigonometric B-splines studied by these authors did not sum up to unity as opposed to polynomial B-splines and thus did not lie in the convex hull of control points.

To get the better of this deﬁciency, Koch et al. (1995) introduced control curves for trigonometric splines instead of control polygon and proved two important properties, which are convex hull property and variation diminishing property for these splines.

The study also includes knot insertion algorithm. It was concluded that by adding extra knots in trigonometric splines, it will lead the control curves in converging the splines.

Certain CAGD application involve curves with such complex shapes that cannot be represented by single curve segments (Buss, 2003). One solution is degree rais- ing that would add ﬂexibility to the curve. However, the processing effort for curve manipulation is considerably increased. In addition, a curve with higher degrees may induce some numerical noise in computations. To cope with this, a curve with complex shape is often represented by stitching together a number of low degree curves (called segments) while meeting certain speciﬁcations of continuity at the joints.

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Such curve is termed as a piecewise curve. A curve that is made of several Bézier curves is called a composite Bézier curve (Farin, 2002a). In order to create a good composite Bézier curve, smoothness is a very important element. Smoothness is a primitive requirement in generating curves. The order of smoothness is application dependent. For example, in architectural design, it is sufﬁcient that the curve segments only bear position continuity. It is not enough to just match the curves exclusively at their adjoining points.

A smooth joint is possible only if some extra conditions are imposed at those points.

Thus, parametric and geometric continuity conditions are used in order to get a smooth joint of two or more pieces of a single curve. Parametric continuity is accomplished by matching the parametric derivatives of adjacent segments of curves at their common boundary (Farin, 2002a). The ﬁrst nth-order parametric continuity where the curve segments meet are known asCⁿ. The three most commonly used levels of parametric continuity are:

• C⁰stands for position continuity. The adjoining curve segments are merely connected at their respective endpoints.

• C¹suggests that the ﬁrst order of parametric derivatives of successive segments at the shared points to be equal. Therefore, the two curve segments have the same tangent at the common point, both in magnitude and direction.

• C² means that both ﬁrst order and second order parametric derivatives of two curve segments are same at the intersection point. Thus,C²denotes the continuity of acceleration vectors.

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A relaxed form of nth-order parametric continuity, Cⁿ has been developed and dubbed nth-order geometric continuity or Gⁿ (Barsky and DeRose, 1989). Geomet- ric continuity depends upon the geometry of the curve (Meek and Walton, 2002).

• G⁰is equivalent toC⁰.

• G¹ requires that the two curve segments and their unit tangent vectors are continuous at the joint.

• G²means that the two adjacent curves areG¹and in addition their signed curvatures at the joining point are continuous.

Curvature formulas for parametrically deﬁned curves and surfaces are well-known both in the classical literature on Differential Geometry (Spivak, 1975; Struik, 2012) and in the contemporary literature on Geometric Modeling (Farin, 2002a; Hoschek et al., 1993). For planar curves, curvature has several equivalent deﬁnitions by Gold- man (2005). The curvature of a curve is the measure of its deviation from a straight line in a neighborhood of a given point, and the curvature becomes greater as this deviation becomes greater.

2.2 Spiral Curves

Spiral curves are generally used to provide a gradual change in curvature from a straight section of road to a curved section (Sipes and Sipes, 2013). Spiral was ﬁrst used in the late 1800s, and its use peaked in the design of the parkways of the 1930s (Myers, 2001). The use of spirals was ﬁrst documented in the late 1600s in

"Sino Loria," by James Bernoulli (Higgins, 1922). Spirals were rediscovered in 1874

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by Cornu (Kimia et al., 2003) and used in optics and later spirals began to replace parabolic curves in easing transitions for railroads.

A transition curve, or easement curve, as it is sometimes called, is a curve of vary- ing radius used to connect circular curves with tangents for the purpose of avoiding the shock and disagreeable lurch of trains, due to the instant change of direction and also to the sudden change from level to inclined track (Talbot, 1904). The term spiral is interchangeable with easement or transition curve. Spirals are curves used to transition between a circular curve with a speciﬁc radius and degree of curvature and a straight tangent. Spiral provides a gradual transition from moving in a straight line to moving in a curve around a point.

A spiral curve is a geometric feature that can be added to a regular circular curve which the radius and sharpness of a spiral curve are increased uniformly along its length (Myers, 2001). Spiral is the most important characteristic in the development of railroad and highway by allowing a safe transition from straight to curved section of track. The use of a spiral is to make roads or tracks to follow the same form that the vehicle naturally takes. The length and degree of curvature of a spiral curve are based on the anticipated speed of trafﬁc and the sharpness of the circular curve that the spiral must meet. In cars, you don’t go directly from going straight to fully turning. There is a transition area where you slowly turn the steering wheel.

Lateral acceleration is slowly increased as the spiral is entered, or it is slowly de- creased as the spiral is exited. Spiral curve enable railroad vehicles to proceed into simple curve without derailing. There are a few advantages of spiral curves. Firstly,

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it provides a natural path such that the lateral force increase and decrease gradually as the vehicle enters and leaves the circular curve. Secondly, the length of the transition curve provides a suitable location for the superelevation runoff. Next, the spiral curve facilitates the transition in width where the travelled way is widened on circular curve.

Lastly, the appearance of the highway is enhanced by the application of spiral curves.

2.3 Design of Highways

Highway is an important aspect when it comes to transportation. Transport plays an important role in our daily activities, such as for economic purposes where raw ma- terials, fuels, food, and manufactured goods are moved from where there are surpluses to where there are shortages. There are various modes of transportation such as air, rail, road, water, cable, pipeline, and space. Basically, it operates using three main ways which are land, water, and air. There are certain factors that needed to be considered when choosing a transport, such as speed, comfort, cost, ﬂexibility, regularity, and safety. Considering the speed, air transports are the quickest mode, but its also the costliest of them all.

A research on safety criteria for highways’ curve design had been done (Anderson et al., 1999; Lamm et al., 1995). The research was conducted to evaluate the validity of design criteria for horizontal highway curves. The evaluation was speciﬁcally con- cerned with the design equation, assumed levels of tire-pavement side friction capabil- ity, safe side friction factors, maximum degree of curvature, maximum superelevation, and design factors of safety. Besides that, Ivan et al. (2009) had done their research on designing roads that guide drivers to choose safer speeds. Their report describes

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an investigation on how actual vehicle running speeds are associated with the occur- rence and severity of motor vehicles crashed in conjunction with roadway and roadside characteristics.

In Malaysia, guideline on geometric design of roads had been published by Jabatan Kerja Raya (1986) (JKR) and Hamzah et al. (1992). These guidelines were applied to all new construction and improvements of roads for vehicular trafﬁc . In JKR’s module, they categorized the area of roads into two categories, which are rural and urban areas.

Then, the speciﬁc value of superelevation rate was applied to those areas in calculating the speed for drivers to operate their vehicles.

For practical purposes in establishing the design criteria for the roads, a maximum superelevation rate of 0.10 is used for roads in rural areas and 0.06 for roads in urban areas. The roads in rural areas are further classiﬁed into ﬁve categories by function namely expressway, highway, primary road, secondary road and minor road, and into four categories in urban areas, namely, expressway, arterial, collector and local street.

Lower design speeds are usually adopted for urban roads to take into account the nature of trafﬁc and adjoining land use. Thus, every categorized area of the road should be connected and consistently designed.

2.3.1 Design Consistency

Consistency of the design can be quantiﬁed by examining speed measures, align- ment measures and workload measures. Consistency of design can be deﬁned as the conformance of a road’s geometric and operational features with driver expectancy (Wooldridge, 2003). Design consistency is a crucial element in designing roads to

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avoid situations where drivers are surprised by unexpected features of the road. Thus, it is important to construct roads with good continuity and less sharp corners to im- prove safety.

Researches were put on the table over the years to develop an efﬁcient methodology. The primary purpose was to remove the element of surprise from driving tasks.

Surprise elements can put drivers under pressure which lead them to making wrong decisions and possibly resulting in losing control of the vehicle (Wolhuter, 1994). Fur- ther researches was made to predict operating speeds on curves of a range of radii.

Various methods have been developed to quantify the consistency of designs in terms of variations in speed along the length of the road or drivers’ workload. It is now possible to assess the consistency of design objectives as opposed to reliance on sub- jective judgement. Good operating speed of the roadway can be assessed by its design consistency.

2.3.2 Design Speed

Design speed was the ﬁrst measure to be applied to the achievement of the consistency of design but has been shown to have weaknesses. A methodology had been developed where consistency can be quantiﬁed (Lamm et al., 1999). Roads are designed according to a speed design which is constant for a given stretch of roadway.

Thus, a vehicle must be able to travel comfortably and safely at the length of a given stretch of road at the design speed regardless of bends. Design speed was deﬁned as the maximum speed at which the road could be traversed safely when only the geometry of the road dictated what this speed could be. A major weakness of design speed as

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a measure of consistency is that it only deﬁnes the minimum geometric standards that could be adopted in the design of roads. Consistency could thus imply that the entire road should have geometric standards that are close to minimum values.

This deﬁnition of consistency had been replaced by the statement that ‘Design speed is the speed selected to determine the various geometric design features of the road for design’ by (AASHTO, 2001). Design speed is thus no longer a speed per se but rather constitutes a grouping of geometric standards. Consistency of design dictates that it should be logical with regard to the topography, adjacent land used, and the functional classiﬁcation of the road, all of which would have a bearing on expected operating speeds.

Vehicle running speed associates with consistency of design of the pathway. Two long tangents connected by a 60 km/h radius curve could quite correctly and yet totally misleadingly be described as having a design speed of 60 km/h. The next development was to use the operating speed from 85 per cent of maximum design speed, as the basis for achieving consistency of design (Fitzpatrick et al., 2003). This new lower speed was adopted for design, albeit brieﬂy. It was then found that, under conditions of long tangents and gentle radii of curvature, operating speed could actually be higher than design speed by anything up to 10 or even 20 km/h. In consequence, operating speed regained respectability as a measure of design consistency.

A successful design will lead to a road that provides safety, accessibility, mobility, and convenience to the movement of people and goods with minimum side effects. A road can never be guaranteed totally safe, but it can be designed to have safety features.

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Usually, crash victims would say that the factors that contributed to the accident were black spots rather than individual error. Black spots occur on roads where every element conforms to standard and suggest that many drivers are making similar mistakes at the same places (Wolhuter, 1994).

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CHAPTER 3 TRIGONOMETRIC BÉZIER CURVE

3.1 Introduction

Curves are commonly used in industry and engineering ﬁeld (Han et al., 2009), while surfaces are mostly being used by medical practitioners to aid them during medical operational process (Iscan and Steyn, 2013). Uses of curves such as B-spline and Bézier are based on their basis that span the entire element. Over the years, other than polynomial space, many basis had been presented in new spaces.

Peña (1997) constructed a basis C_m =span{1,cost,....,cosmt} where m∈ Z⁺. Curves in the space span of{1,cost,sint,....,cosmt,sinmt}have been developed by Zhang (1996). Ahmad et al. (2014) constructed a new generalization basis called Bézier-like curve. However, Mainar et al. (2001) also constructed α-basis of quartic Bernstein. The other approach by Chen and Wang (2003), used integral as their basis called C-Bézier.

Most of these bases do not enclose free form curves of higher-order polynomial.

Recently, the degree of trigonometric Bézier curve expanded from quadratic by Bashir et al. (2012), cubic trigonometric Bézier curve with shape parameter by (Han et al., 2009), quartic (Dube and Sharma, 2013; Zhu et al., 2012), quasi-quintic (Bashir et al., 2013), and quintic (Dube and Yadav, 2014).

In this chapter, we present a new basis of quintic trigonometric Bézier curve with

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two shape parameters. This new basis provides another alternative curve compared to Dube and Yadav (2014). Dube and Yadav (2014) used four control points that mimic cubic trigonometric Bézier curves with a single shape parameter. Here, six control points with two shape parameters and some properties are discussed. Later, we will concentrate on how different parameters affect those types of curve.

3.2 Cubic Trigonometric Bézier Curve

Cubic trigonometric Bézier curve with two shape parameters had been proposed by Han et al. (2009). The two shape parameters allow designers to change the curve according to their desired shape while keeping the control polygon unchanged. More- over, this unique function can also approximate ellipses and arc segments of circles.

3.2.1 Cubic Trigonometric Bézier Basis Function

Cubic trigonometric Bézier curve with two shape parameters from Han et al. (2009) is given by:

z(t) =

∑

³

i=0

P_if_i(t), t ∈[0,1] (3.1)

whereP_iare the control points and f_iare the cubic trigonometric basis functions for all i=0,1,2,3 with values of shape parameters p,q∈[−2,1]. The f_iare deﬁned as:

f₀(t) = (1−sinπt

2)²(1−psinπt 2), f₁(t) =sinπt

2 (1−sinπt

2 )(2+p−psinπt 2 ), f₂(t) =cosπt

2 (1−cosπt

2 )(2+q−qcosπt 2 ), f₃(t) = (1−cosπt

2)²(1−qcosπt 2 ).

(3.2)

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Figure 3.1 shows plots of cubic trigonometric Bézier basis function for two arbi- trarily selected real values of p and q, where p,q∈[−2,1]. In Figure 3.1, different value of shape parameters give different projections of the basis function as such red- dotted curve (p=−2,q=−2), blue-dotted curve(p=−1,q=−1), purple-dashed curve(p=0,q=0)whereas black-line curve(p=1,q=1).

Figure 3.1: Cubic trigonometric Bézier basis function with different values of shape parameters

Cubic trigonometric Bézier basis function in Equation (3.2) have the following properties:

i. Non-negativity: f_i t

≥0, fori=0,1,2,3.

ii. Partition of unity:∑³_i=0 f_i(t)≡1.

iii. Symmetry: f_i(t;p,q) = f_3−i(1−t;p,q)for everyi=0,1,2,3.

The proof for properties of cubic trigonometric Bézier can be obtained in Han et al.

(2009).

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3.2.2 Properties of Cubic Trigonometric Bézier Curve

Cubic trigonometric Bézier curve in Equation (3.1) has the following properties (Han et al., 2009):

(a) Endpoint terminal

The endpoint terminal for cubic trigonometric Bézier curve are provided in Equation (3.3) until Equation (3.5):

z(0) = P₀ z(1) = P₃

(3.3)

z(0) = ^π₂(P₁−P₀)(2+p) z(1) = ^π₂(P₃−P₂)(2+q)

(3.4)

z(0) = ^π₂²[(2p+1)(P₁−P₀) + (P₂−P₁)]

z(1) = ^π₂²[(2q+1)(P₃−P₂) + (P₂−P₁)]

(3.5)

(b) Convex hull

The entire cubic trigonometric Bézier curve segment must lie inside its convex hull spanned byP₀,P₁,P₂,P₃.

(c) Symmetry

P₀,P₁,P₂,P₃andP₃,P₂,P₁,P₀deﬁne the same trigonometric Bézier curve in different parametrization, i.e.,z(t;p,q:P₀,P₁,P₂,P₃) =z(1−t;p,q:P₃,P₂,P₁,P₀) where 0≤t≤1 and p,q∈[−2,1]if and only if p=q.

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(d) Geometric invariance

Geometric invariance for cubic trigonometric Bézier curve are as follows:

z(t;p,q:P₀+m,P₁+m,P₂+m,P₃+m) =z(t;p,q:P₀,P₁,P₂,P₃) +m z(t;p,q:P₀×n,P₁×n,P₂×n,P₃×n) =z(t;p,q:P₀,P₁,P₂,P₃)×n

where 0≤t≤1, p,q∈[−2,1],mis an arbitrary vector inR²orR³, andnis an arbitraryd×dmatrix withd=3 or 4.

3.3 Quintic Trigonometric Bézier Curve

In this section, a new type of trigonometric function called quintic trigonometric Bézier curve with two shape parameters is proposed. This function consists of six basis functions and possesses some of the properties that will be discussed below.

3.3.1 Quintic Trigonometric Bézier Basis Function

Quintic trigonometric Bézier curve with two shape parameters are given as follows:

z(t) =_i=0

∑

⁵ ^Pⁱ^gⁱ⁽^t^), ^t^∈^[0,^1] ^(3.6)

wherePiare the control points andgi are the quintic trigonometric basis functions for alli=0,1,2,3,4,5 with values of shape parameters p,q∈[−4,1]. Theg_i are deﬁned as:

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g₀(t) = (1−sinπt

2 )⁴(1−psinπt 2 ), g₁(t) =sinπt

2 (1−sinπt

2)³(4+p−psinπt 2 ), g₂(t) = (1−sinπt

2 )²(1−cosπt

2 )(8 sinπt

2 +3 cosπt 2 +9), g₃(t) = (1−cosπt

2 )²(1−sinπt

2 )(8 cosπt

2 +3 sinπt 2 +9), g₄(t) =cosπt

2 (1−cosπt

2)³(4+q−qcosπt 2 ), g₅(t) = (1−cosπt

2 )⁴(1−qcosπt 2 ).

(3.7)

Quintic trigonometric Bézier basis function have the following properties:

i. Non-negativity:g_i t

≥0, fori=0,1,2,3,4,5. ii. Partition of unity:∑⁵_i=0g_i(t)≡1

iii. Symmetry:g_i(t;p,q) =g_5−i(1−t;p,q), ifp=qfori=0,1,2,3,4,5. Proof.

i. Fort∈[0,1]andp,q∈[−4,1], then 1−psin^πt₂ ≥0, 1−qcos^πt₂ ≥0, 1−sin^πt₂ ≥0, 1−cos^πt₂ ≥0, sin^πt₂ ≥0, cos^πt₂ ≥0, 4+p−psin^πt₂ ≥0, 4+q−qcos^πt₂ ≥0.

Therefore,

g₀= (1−sin^πt₂)⁴(1−psin^π₂^t)≥0,

g₁=sin^πt₂(1−sin^πt₂)³(4+p−psin^πt₂)≥0,

g₂= (1−sin^πt₂)²(1−cos^πt₂)(8 sin^πt₂ +3 cos^π₂^t+9)≥0, g₃= (1−cos^π₂^t)²(1−sin^πt₂)(8 cos^πt₂ +3 sin^π₂^t+9)≥0, g₄=cos^πt₂(1−cos^πt₂)³(4+q−qcos^πt₂)≥0,

g₅= (1−cos^π₂^t)⁴(1−qcos^πt₂)≥0.

TRIGONOMETRIC BEZIER TRANSITION CURVES AND APPLICATIONS

TRIGONOMETRIC BEZIER TRANSITION CURVES AND APPLICATIONS

MD YUSHALIFY MISRO

UNIVERSITI SAINS MALAYSIA

2017

TRIGONOMETRIC BEZIER TRANSITION CURVES AND APPLICATIONS

MD YUSHALIFY MISRO

August 2017

ACKNOWLEDGEMENT

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF ABBREVIATIONS

LIST OF SYMBOLS

CHAPTER 1 INTRODUCTION

CHAPTER 2

BACKGROUND AND LITERATURE REVIEW

CHAPTER 3

TRIGONOMETRIC BÉZIER CURVE

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