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Construction of Quintic Trigonometric Bézier Spiral Curve ASM Sc. J., 12, Special Issue 5, 2019 for ICoAIMS2019, 208-215 Construction of Quintic Trigonometric Bézier Spiral Curve M.Y. Misro1∗, A. Ramli1 and J.M. Ali1 1School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Gelugor, Pulau Pinang, Malaysia The construction of curves need to satisfy some conditions in order to achieve smooth pathways. In terms of designing highways, there are five design templates of transition curves. Usually, cubic and quintic Bézier curves are used to construct those five templates. However, quintic trigonometric Bézier curves can also be used to construct these five templates of transition curves. Global and local dynamic parameters that are embedded in control points of quintic trigonometric Bézier will provide more freedom for designers to comply with difficult tasks in designing. Keywords: spiral; smooth pathway; quintic curve; trigonometric Bézier; dynamic parameters I. INTRODUCTION Spiral curves consist of several types beginning with Euler Spiral, Logarithmic Spiral, Parabolic Spiral, and Spiral curves can be found from the representation of growth more. Euler spiral is a beautiful and useful curve known and the process of the Nautilus shell, to the patterns of by several other names, including clothoid and Cornu sunflower seeds, to the design of a staircase in the Vatican spiral (Levien, 2008). One unique property of this spiral Museum, and to the method of construction of curve is that the curvature increases linearly with its railway/roadway design. A spiral curve is one of the features arclength. Levien (2018) stated that the problem of in geometric field especially in Computer Aided Geometric elasticity were the first appearance of Euler Spiral Design. This important feature can be added on to a regular published by James Bernoulli in 1694. circular curve. The spiral can be defined as a curve segment The logarithmic, or equiangular spiral was first that provides a gradual transition between a straight line to a discovered by Rene Descartes (1596-1650) in 1637 by circle. In engineering, spirals are used to overcome the abrupt representing various curves with the help of equations. change in curvature and superelevation that occurs between Logarithmic spiral was one of those curves which at that tangent and circular curve especially for moving objects time drew the attention of mathematicians. At present, (Misro et al., 2015). that equation is written in the form 푟 = 푒훼휃. A parabolic The study of the natural structures will help numerous spiral can also be represented by the mathematical designers, architects, or even engineers to learn the secrets of equation 푟2 = 푎2휃. This spiral discovered by Bonaventura each natural form. Archetypal motif in spiral curve itself Cavalieri (1598-1647) creates a curve commonly known as brings its own uniqueness that can be represented in various a parabolic spiral (Mukhopadhyay, 2004). forms. Spirals that consist of strength, versatility, and Recently, the study of spiral construction using any kind pleasing looks will be the indicator for designers. The study of of method has aspired many researchers, including the the natural spiral form is inclusive of the features of human use of trigonometric Bézier curve that comprises the face that were derived by using golden spiral, whereby the flexibility of dynamic parameters. Previously, (Misro et al., ratio represented by the spiral constant is the only growth 2017) had published his work about spiral curve using ratio by which a new unit can grow in proportion to the old cubic trigonometric Bézier curve. This research proved unit; and still retain the same shape. This is exactly the same that spiral curves can be constructed using trigonometric process of growth in a spiral shell. function and not limited to only polynomial and hyperbolic function (logarithmic spiral). *Corresponding author’s e-mail: [email protected] ASM Science Journal, Volume 12, Special Issue 5, 2019 for ICoAIMS2019 In this research, a method to construct a spiral curve by 1 2 2 푃3 = 푃2 + (2푢1 − 2푣1 + 푢0푢2 − 푣0푣2, 4푢1푣1 increasing the number of segments will be proposed. 15 + 푢 푣 + 푢 푣 ), Therefore, a higher degree of curve is required in order to 0 2 2 0 1 bring more flexibility to the spiral curve. Section 2 briefly 푃 = 푃 + (푢 푢 − 푣 푣 , 푢 푣 + 푢 푣 ), 4 3 5 1 2 1 2 1 2 2 1 explains the background notations used throughout this 1 2 2 푃5 = 푃4 + (푢2 − 푣2 , 2푢2푣2), (3) research including the basis function of quintic trigonometric 5 Bézier curve. Section 3 summarises all the conditions of the Pythagorean Hodograph (PH) quintic trigonometric Bézier where 푃0 is arbitrarily chosen. PH curve can be defined in spiral curve. Another four templates of transition curves will parametric form as be demonstrated in Section 4 to 7. Some remarks and suggestions for future works are concluded in Section 8. 푧′(푡) = (푢(푡)2 − 푣(푡)2, 2푢(푡)푣(푡)), 0 ≤ 푡 ≤ 1 (4) Where 2 2 II. NOTATION AND 푢(푡) = 푢0(1 − 푡) + 2푢1(1 − 푡)푡 + 푢2푡 , CONVENTION 푣(푡) = 푣 (1 − 푡)2 + 2푣 (1 − 푡)푡 + 푣 푡2. (5) 0 1 2 Quintic Trigonometric Bézier curve with local dynamic Its signed curvature 휅(푡) is given by parameters 푝 and 푞 proposed by (Misro et al., 2017b) will be used throughout this paper, and given as follows 푧′(푡)× 푧′′(푡) 2(푢(푡)푣′(푡)−푢′(푡)푣(푡)) 휅(푡) = ′ 3 = 2 2 2 (6) ‖푧 (푡)‖ (푢(푡) +푣(푡) ) 5 푧 (푡) = ∑푖=0 푃푖푔푖(푡) (1) III. QUINTIC TRIGONOMETRIC where 푃푖 are control points and 푔푖 be the quintic trigonometric BÉZIER SPIRAL CURVE basis function for every 푖 = 0, 1, 2, 3, 4, 5 which are defined as Apart from cubic Bézier curve, PH quintic Bézier curve is 휋 푡 휋 푡 oftenly used as a basis function to construct transition 푓 (푡) = (1 − 푠푖푛 )4 (1 − 푝 푠푖푛 ), 0 2 2 curve (Habib & Sakai, 2005; Walton & Meek, 2004). In 휋 푡 휋 푡 휋 푡 푓 (푡) = 푠푖푛 (1 − 푠푖푛 )3 (4 + 푝 − 푝 푠푖푛 ), 1 2 2 2 (Misro et al., 2017c), the authors only constructed PH 휋 푡 2 휋 푡 휋 푡 휋 푡 푓 (푡) = (1 − 푠푖푛 ) (1 − 푐표푠 ) (8 푠푖푛 + 3 푐표푠 + 9), quintic trigonometric Bézier transition curve. Therefore, in 2 2 2 2 2 휋 푡 2 휋 푡 휋 푡 휋 푡 this paper, we will construct spiral curves using quintic 푓3(푡) = (1 − 푐표푠 ) (1 − 푠푖푛 ) (8 푐표푠 + 3 푠푖푛 + 9), 2 2 2 2 trigonometric Bézier with the following conditions to be 휋 푡 휋 푡 휋 푡 푓 (푡) = 푐표푠 (1 − 푐표푠 )3 (4 + 푞 − 푞 푐표푠 ), 4 2 2 2 satisfied 휋 푡 휋 푡 푓 (푡) = (1 − 푐표푠 )4(1 − 푞 푐표푠 ). (2) 5 2 2 1 Local dynamic parameters of 푝 and 푞 in Equation (2) will 푧(0) = (0, 0), (휅(0), 휅(1)) = (0, ), (7) 푟 ( ) enable the curve to become more flexible. The curve 푧 푡 = 푧′(0) ∥ (1,0), 푧′(1) ∥ (cos 휃 , sin 휃) (8) (푥(푡), 푦(푡)) is said to be a PH curve if 푥′(푡)2 +푦′(푡)2 can be 휋 휃 ∈ (0, ) is an angle that is measured anti-clockwise from expressed as the square of polynomial in 푡. By using the result 2 ′ ′ given by (Farouki & Sakkalis, 1990), control points 푃푖 of the 푧 (0) to 푧 (1). Parameter 푟 is referred as the radius of the quintic trigonometric Bézier curve are as follows circle. Taking quintic trigonometric Bézier curve in Equation (1) with basis function in Equation (2), our initial 1 starting point will be at 푧(0) = (0, 0) in Equation (7). To 푃 = 푃 + (푢2 − 푣2, 2푢 푣 ), 1 0 5 0 0 0 0 have a spiral curve from a straight line to a circle, we need 1 푃 = 푃 + (푢 푢 − 푣 푣 , 푢 푣 + 푢 푣 ), 1 2 1 5 0 1 0 1 0 1 1 0 휅(0) = 0 and 휅(1) = . Its curvature begins with zero at the 푟 straight line and increases linearly with its curve length. Both initial value and end value of the curvature are 209 ASM Science Journal, Volume 12, Special Issue 5, 2019 for ICoAIMS2019 crucial for this spiral curve to ensure a smooth continuity at varies from −4 to 1. the joints. Then, parallel equation in Equation (8) needs to be satisfied before we can obtain the following unknowns such as 푑3 푑3 휃 (푢 , 푢 , 푢 ) = (푚 , , 푑 cos ), 0 1 2 휃 휃 4푟 sin 4푟 sin 2 2 2 휃 (푣 , 푣 , 푣 ) = (0,0, 푑 sin ) (9) 0 1 2 2 with positive parameters 푢0 = 푚푢1 and 푑 = √푠푟 whereas 푚 and 푠 are free parameters to alter the shape of the curve. The Figure 2. Curvature profile of spiral quintic parameter 푚 and 푢0 act as global dynamic parameters that trigonometric Bézier curve will depend on the value of 휃. Thus, quintic spiral curves 푧(푡) that are satisfying the conditions in Equation (7) and Equation (8) can be found. Therefore, the quintic trigonometric Bézier spiral curve can be written a 5 푧(푡) = ∑푖=0 푃푖푔푖(푡) = 푃0푔0(푡) + 푃1푔1(푡) + 푃2푔2(푡) + 푃3푔3(푡) + 푃4푔4(푡) + 푃5푔5(푡) (10) with 푃 = 1, 2, 3, 4, 5 as in Equation (3). 푖 Figure 3. Curvature derivative of spiral quintic trigonometric Bézier curve Curvature profile in Figure 2 is almost linearly increasing until 푡 = 0.85, where it starts to have a flat value until 푡 = 1 . This shows that the curve is entering the circular curve of the circle at the end of the curvature.
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