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Generalization of Approximation of Planar Spiral Segments by Arc Splines A thesis submitted to the faculty of graduate studies in partial fulfillment of the requirements for the degree of Master of Science Department of Computer Science University of Manitoba Winnipeg, Manitoba, Canada 1998 National Library Bibbthéque nationale I*m of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services seMces bibliographiques 395 Wellington Street 395. rue Wellington OttawaON KlAW Ottawa ON K1A ON4 Canada CaMda The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Lîbrary of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distri'buer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'azteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de ceUe-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation, FACULN OF GRADUATE STZI'DIES ***** COPYRIGHT PER%LISSIOXPAGE A ThesidPncticom submitted to the Faculty of Graduate Studies of The University of bfmitoba in partial hilfillment of the requirements of the degree of MASTER OF SC~CR Permiision has been grnnteà to the Library of The University of Manitoba to lend or seii copies of this thesis/practicum, to the National Library of Canada ta microfiIm this thesis and to lend or sel1 copia of the fw,and to Dissertations Abstncts International to pubüsh an abstract of this thesidpracticum. The author reserves other publication righb, and neither this thesiJpracticum nor extensive estracts from it may be printed or otherwise reproduced without the author's written petmission. Abstract Spirals based on quadratic Bézier, cubic Bésier, Pythagorean hodogragh (PH) cu- bic. PH quintic and clothoid curves are suitable /or CAD and cornputer-aided geornet- ric design (CAGDJ applications. The cfothoidal spiral segments are widely used in highwa y design. railwa y design and robot trajecto ries. For CNC machining eompared with polyline approximations, the suggested arc spline approximations avoid sudden changes in the direction of the tool path, decrease the number of segments for approx- imation and lessen the need to polish objects. In this dissertation an existing method is generalized to approximate a planar spiral segment so that it can be applied to a large class of spiral segments. The properties of several spiral segments are analyred and their approzimations by the proposed method are presented. Acknow ledgements I would like to thank rny advisor Dr. Desmond Walton who has provided me uiith guidance, advice, encouragement, and support throughout rn y Master st udies. 1 wish to thank the rnembers of rny cornmittee, Dr. D. Meek and Dr. W. Hoskins /or reading the thesis and oflering invaluable advice to me. 1 would like to take this opportunitg to give a special thanks to my dear friends. I would like to thank my parents. sisters for their constant loue and support. Contents 1 Introduction 1 1.1 Notation. Conventions and Terminology ................ 3 1.2 Literature Review ............................. 11 1.3 Approximation of Spiral Segments .................... 16 2 Theoretical Background 19 2.1 Pyt hagorean Hodograp h Curves ..................... 19 2.1.1 Introduction of Pyt hagorean Hodogragh ............ 20 2.1 Fundamentals of Pyt hagorean Hodographs ........... 21 2.1.3 Pythagorean Hodograph Cubic Bézier .............. 93-_ 2.1.4 Pythagorean Hodograph Quintic Bézier ............. 23 2.2 Biarc .................................... 24 2.3 Quadratic Bézier Spiral .......................... 25 2.4 Cubic Bézier Spiral ............................ 28 2.5 PH Cubic Spiral .............................. 30 2.6 PH Quintic Spiral ............................. 33 2.7 ClothoidalSpiral ............................. 35 ... 111 i v C'OiVTE-VTS 2 Definition ............................. 3.5 2.7. Properties ............................. 36 2-73 Usage ............................... 37 2.7.4 Highway Spiral and the Clothoid ................ 43 2.8 Archimedean Spiral ............................ -!5 2.9 Logarithmic Spiral ............................ -45 2.9.1 Definition and Equiangular Property .............. 4.5 29 Transformation .......................... AS 2.9.3 Reiationship between Logarithmic Spiral and Archimedean Spiral 49 2.9.4 .4 pplications of Logarithmic Spiral and Archimedean Spiral . 51 3 Generalization 53 3.1 Generalized T heorem ........................... 5 .3 3.2Location of Local Maxima ........................ 59 3.2.1 Polynomial Spiral Segment .................... 59 3.2.2 Other Spiral Segments ...................... 61 4 Spirals 65 4.1 Quadratic Bézier Spiral .......................... 66 4.1 Cubic Bézier Spiral ............................ 67 4.3 PH Cubic Spiral .............................. 6s 4.4 PH Quintic Spiral ............................. 69 4.5 Clothoid .................................. 72 4.6 Archimedean Spiral ............................ 72 4.7 Logarithmic Spiral ............................ 73 CONTENTS v 5 Algorithrns 75 -- 1 Algorithm . ri 6 Examples 87 7 ConcIusions 103 7.1 Tendency of the Radius . 103 7.2 Relationship between Number of Arcs and Tolerance . 104 List of Figures 1.1 Inflection Point .............................. 6 2 Curve with Inflection Point ....................... 10 1.3 Curve without Infiection Point ...................... 10 1.4 General Space Curve ........................... 11 1.5 Bounding Circular Arcs ......................... 14 2.1 Biarc .................................... 25 2.2 Quadratic Bézier Spiral Segment .................... 26 2.3 Regular Cubic Bézier Spiral Segment .................. 30 2.4 PH Cubic Spiral Segment ........................ 31 2.5 PH Cubic Tangent at t = 1 ....................... 32 2.6 PH Quintic Spiral Segment ........................ 35 2.7 Clothoidal Spiral ............................. 37 2.8 Circular Arc without Transition ..................... 38 2.9 Compound Circular Arc ......................... 39 2.10 Circular Arc with Spiral Transition ................... ?O 2.1 I Transition Curve ............................. 41 2.12 Archimedean spiral with m = 2 ..................... 46 vii ... ~111 LIST OF FIGURES 2.13 Logarithmic Spiral ............................ 47 2.14 Logarithmic and Archimedean Spiral .................. 50 3.1 Spiral Triangle .............................. 54 3.2 Biarc Approximation of Spiral Segment ................ 55 3 PIotofgi(t) ................................ 62 3.4 PIotofgi(t) ................................ 62 3.5 Bumps Appear ............................... 63 1 Circle Determined by an Isosceles Triangle ............... 76 5.2 incentre of Triangle ............................ 76 5.3 Triangle Construction ........................... 79 6.1 Arc Spline Approximation of Quadratic Bézier Spiral Segment .... 92 6.2 Arc Spline Approximation of Cubic Bézier Spiral Segment ...... 92 6.3 Arc Spline Approximation of PH Cubic Spiral Segment ........ 92 6.4 Arc Spline Approximation of PH Quintic Spiral Segment ....... 93 6.3 Arc Spline Approximation of Clot hoid Segment ............ 93 6.6 Arc Spline Approximation of Archimedean Segment .......... 93 6.7 Arc Spline Approximation of Logarithmic Spiral Segment ...... 94 6.8 QuadraticBézierSpiral .......................... 99 6.9 Cubic Bézier Spiral ............................ 99 6.10 PH Cubic Bézier Spiral .......................... 100 6.1 1 PH Quintic Bézier Spiral ......................... 100 6.12 Clothoidal Spiral ............................. 101 6.13 Logarithmic Spiral ............................ 101 LIST OF FIGURES 6.14 Archimedean Spiral . 7.1 Logarithmic Pact of Deviation vs. No. of Arcs . 105 List of Tables 6.1 Results of Approximation of Quadratic Bézier Spiral Segment . 6 Results of Approximation of Cubic Bézier Spiral Segment . 6.3 Results of -4pproximation of PH Cubic Spiral Segment . 6.4 Results of Approximation of PH Quintic Bézier Spiral . 6 Results of Approximation of Clothoid Spiral Segment . 6.6 Results of Approximation of Archimedean Spiral Segment. 6.7 Results of Approximation of Logarithmic Spiral Segment . 6.8 Radius of Approximation of Quadratic Bézier Spiral Segment by Arc Spline ................................... 6.9 Radius of Approximation of Cubic Bézier Spiral Segment by Arc Spline 6.10 Radius of Approximation of PH Cubic Spiral Segment by Arc Spline . 6.11 Radius of Approsimation of PH Quintic Spiral Segment by Arc Spline 6.12 Radius of Approximation of Clothoid Spiral Segment by Arc Spline . 6.13 Radius of Approximation of Logarithmic Spiral Segment by Arc Spline 6.14 Radius of Approximation of Archimedean Spiral Segment by Arc Spline 98 Chapter 1 Introduction With the influence of automatic control of machines and cornputer-aided design (CAD), computer-aided manufacturing (CAM) systems have expanded in manufac- turing industries during the last two or three decades. Designers use CAD systems to design parts for visual and theoretical analysis. On the basis of the CXD model. the numerical control (NC) programmer uses the C.Ab1 system to generate a NC tool path for a computerized numerical
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