Review on the Curves and Surfaces Modeling Technology

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1 IOSR Journal of Mechancal and Cvl Engneerng (IOSR-JMCE) e-issn: ,p-ISSN: X, Volume 15, Issue 6 Ver. II (Nov. - Dec. 018), PP WuYze 1, Zhang Xu 1, Qu Kangjan 1, Chen Yong 1, Jang Mngyang 1 1 (School of Mechancal and Automotve Engneerng, Shangha Unversty of Engneerng Scence, Chna) Correspondng Author: WuYze Abstract: Wth the contnuous development of computer technology, curve and surface modellng technology s more and more wdely used n modern ndustral product desgn and manufacturng process. However, due to errors n desgn and measurement, the smoothness of curves often fals to meet the requrements of desgn and manufacture. Curve smoothness not only affects the appearance of the product, but also drectly affects the degree of dffculty n manufacturng process and the mechancal propertes of the product. It s an mportant sgn to evaluate the qualty of the product. The research status of curve and surface modellng technology, nterpolaton, approxmaton and fttng of curves, farng crtera and farng methods of curves are dscussed. Keyword- Curve and surface modelng; Curve farng; nterpolaton; approxmaton; fttng Date of Submsson: Date of acceptance: I. Introducton Wth the development of computer technology, especally mcrocomputers and ther drawng technology, Computer Aded Desgn (CAD) and Computer Aded Manufacturng (CAM) have been wdely used n machnery, electroncs, aerospace, constructon and other felds. applcaton. The development and applcaton level of CAD/CAM technology has gradually become one of the mportant ndcators to measure the level of modernzaton of a country. Snce CAD/CAM technology orgnated n the avaton ndustry, due to the complex shape of the arcraft and the large number of free-form curved surfaces, CAD/CAM technology s closely lnked to the freeform surface modelng technology from the begnnng. The curved surface modelng technque was establshed by Coons, Bezer and other masters n the 1960s[1]. After years of research and development, the curve and surface modelng technology has formed a geometrc theory system wth the NURBS curve and surface parameterzed feature desgn and mplct algebrac curve and surface representaton as the man body, wth nterpolaton, approxmaton and fttng as the skeleton. II. Research status of curve and surface modelng technology.1 Curve surface modelng technology classfcaton The contnuous expanson of CAD/CAM technology applcatons has led to an ncreasng demand for curved surface modelng. Because the tradtonal curve and surface modelng method s too smple, t s not convenent for the user to use, and t s dffcult to effectvely modfy the curved surface, some newer and more flexble curved surface modelng methods have appeared, as follows: (1) Functon curve surface modelng technology () Parametrc curve surface modelng technology (3) Implct curve and surface modelng technology (4) Curved surface modelng technology based on partal dfferental equaton (5) Curved surface deformaton method modelng technology (6) Subdvson curve surface modelng technology (7) Curved surface modelng technology based on physcal model (8) Curved surface modelng technology of shape mxng (9) Other modelng technques. Development hstory of curve and surface modelng technology As the basc geometrc element of the CAGD system, the man form of expresson has experenced the followng development stages n the past few decades: In 1963, Ferguson frst proposed a vector functon method that represented curves and surfaces as parameters, and ntroduced the parameters cubc and bcubc surface patches[][3].the parametrc form of curve and surface becomes the standard form of shape mathematcal descrpton from then on. Before that, the descrpton of the curve has always been n the form of explct scalar functon y y x or mplct functon DOI: / Page

2 equaton F x, y 0, and the descrpton of the surface s n the standard form of z z x, y or F x, y, z 0. In 1964, Coons[4] proposed a general surface descrpton method that defnes a surface by gvng four boundares around a closed curve.although such a surface s smple n form, there s an nconvenence n shape control and splcng.in 1967, Coons[5] further promoted ths dea. But both have shape control and connectvty ssues. In 197, Bezer[6] proposed a new method for controllng the desgn curve of a polygon.curved surfaces generated by Bezer technology have a number of good propertes, such as convex hull, affne transformaton nvarance, varablty reducton, and endpont nterpolaton. The method s not only smple and easy to use, but also solves the overall shape control problem beautfully, and advances the desgn of the curved surface to a bg step, layng a sold foundaton for the further development of the curved surface modelng. but there are stll connecton problems and partal modfcaton problems. In 197, Boor [7] and Cox [8] each ndependently proposed a standard algorthm for B-splne calculaton, whch made B-splne begn to be wdely used.the B-splne method almost nherts all the advantages of the Bezer method, overcomes ts shortcomngs, and successfully solves the local control problem and the splcng problem on the bass of contnuous parameters. Wth the technology of node nserton technology, segmentaton technology and ascendng order technology, the B-splne method has entered a more practcal stage, and theoretcally proved that Bezer curve and surface s a specal case of B-splne curve and surface. In 1974, Gordon and Resenfeld appled B-splne theory to shape descrpton and proposed a B-splne curve and surface method[9-10]. Ths method almost nherts all the advantages of the Bezer method, overcomes the shortcomngs of the Bezer method, successfully solves the local control problem, and easly solves the sttchng problem on the bass of parameter contnuty, thus makng the shape of the free curve surface. The descrpton problem has been better solved. Wth the development of producton, the B-splne method shows obvous defcency: t cannot accurately represent the conc secton lne and the elementary analytc surface, whch results n the unque geometrc defnton of the product, so that the curve and surface have no unfed mathematcal descrpton, whch s easy to cause confuson n producton management. In 1975, Versprlle[11]frst proposed a ratonal B-splne method, proposed the NURBS method, and studed ts propertes. Ths method can not only express free-form surfaces, but also accurately represent conc curves, quadrc surfaces and rotatng surfaces. Snce then, untl the late 1980s, the NURBS method has fnally become the most popular mathematcal method n modern curved surface modelng manly due to the achevements of Pegl[1-13] and Tller[14-15]. In 1991, the Internatonal Standardzaton Organzaton (ISO) promulgated an nternatonal standard for Standard for Exchange of Product model data(step), usng the NURBS method as the only mathematcal method for defnng the geometry of ndustral products, makng the NURBS method the most mportant foundaton for the development of curved surface modelng technology. III. Interpolaton, approxmaton and fttng of curves 3.1 Interpolaton What s nterpolaton? Interpolaton, n short, s to construct approxmate functon functon values of some ponts of gven unknown functon x of f f x, whch s called nterpolaton functon x x wth the. 3.Approxmaton What s approxmaton? Approxmaton, smply speakng, constructng a curve to make t most close to a set of gven type ponts n a certan sense, we call t curve approxmaton, the constructed curve s called approxmaton curve. 3.3 Fttng Interpolaton and approxmaton are collectvely referred to as fttng[16]. Fttng does not have a complete defnton and mathematcal expresson lke the nterpolaton and approxmaton mentoned above. Fttng means that n the desgn process, the generated curves can meet some desgn requrements by nterpolaton or approxmaton, such as "smooth" and "smooth" curves. For curves, smoothng refers to ther contnuty or more precse requrements on tangent vectors. IV. Curve smoothng 4.1 Farng meanng What s "farng"? "Farng", as the name suggests, s smooth and pleasng to the eye. "Smoothness" and what we call "smoothness" n our daly lfe are two dfferent concepts, whch cannot be confused. "Smoothness" usually refers to the parametrc contnuty or geometrc contnuty of a curve. It s a mathematcal term. And "smoothng" has not only the requrement of contnuty n mathematcs, but also the requrement of DOI: / Page

3 functon (such as aesthetcs, mechancs, NC machnng). Because t nvolves the aesthetcs of geometrc shape and s nfluenced to a great extent by people's subjectve factors, n general, "smoothness" s stll a vague concept wthout an accurate defnton and unfed standard. Now, can we not smooth the curve? The answer s No. If there s no nherent rule for smoothng, how can we judge the smoothness of curves? Thus, the key to the problem s not whether farness has objectvty, but how to coordnate the relatonshp between the objectvty and uncertanty of farness, whch s the problem to be solved by the defnton and crteron of farness. Two problems need to be solved for smoothng curves: (1) what knd of curve s smooth, that s, farng crtera; () For unfared curves, whch mathematcal treatment should be adopted to satsfy or mprove ther farness, that s, farng treatment method. 4.Curve farng crtera To smooth curves, we need to gve specfc farng crtera frst. Here are some farng crtera whch are often used n farng processng. (1) For plane curves farng crtera1(farn proposed)[17]: For a curve, f ts correspondng curvature curve s contnuous, has approprate symbols (f the concavty and convexty of the curve are known), and s as close as possble to a pecewse monotone functon wth as few monotone segments as possble, then the curve s consdered monotone. farng crtera(su Buqng,Lu Dngyuan proposed)[18]: (a) The two order parameter s contnuous (C contnuous). (b) There are no addtonal nflecton ponts. (c) The curvature changes are more unform. farng crtera3(sh Fazhong proposed) [19]: (a) The two order geometrc contnuum (refers to the poston, tangent drecton and curvature vector contnuous, as G). (b) There are no sngular ponts and unnecessary nflecton ponts. (c) The curvature changes more evenly. (d) Stran energy s small. () For space curve The followng crtera are proposed n document[0]: () Two order smoothness (a) The two order vector of a curve s contnuous, and the curvature s contnuous. (b) The curve (quadratc) of the low order splne may have a jump n the curvature of the node, whch requres the jump degree and the mnmum possble. k t k t (1) () There s no excess nflecton pont (a)there should be G nflecton ponts n the curve, whle there are more than G nflecton ponts n fttng (nterpolaton and approxmaton). (b) There should be an nflecton pont where there should be no turnng pont. () The curvature changes more evenly (v) There s no redundant varable deflecton pont (pont whose deflecton s zero, usually related to the pont whose deflecton s varable), that s to say, the followng stuaton s not allowed: (a) There should be H deflecton ponts, and there are redundant H deflecton ponts when fttng (nterpolaton and approxmaton). (b) There should be no turnng pont where there should be no turnng pont. (v) Unform varaton of torson (a) The torson may be dscontnuous at the node, and the leapng and small enough should be made at ths tme. t t () (b) The varaton of torson s unform, and there s no contnuous sgn change. 4.3Curve smoothng method At present, accordng to the number of type ponts (or control ponts) modfed each tme, curve farng algorthms can be dvded nto global farng and local farng. The global farng algorthm s represented by least squares method and energy method. Ths knd of farng algorthm can reduce the curvature of the curve as a whole and make the curvature of the curve change more unformly, but the shape of the fared curve changes greatly compared wth the orgnal curve, and the drecton of deformaton cannot be controlled. In the local DOI: / Page

4 farng algorthm, the pont selecton modfcaton method and curvature method are used. Rate method s representatve. Ths knd of smoothng method has better smoothng effect n a small local range, but t cannot elmnate the concave defects. (1) Global farng (a) least square method In curve reconstructon, least squares method s the most mportant method. Its basc prncple s to consder the devaton of approxmate functon x x y 0,1, L, m on gven value pont x, y 0,1,, m 1. Mnmze x from L as a whole, and to make t the smallest accordng to a certan measure standard. y 0max m. Mnmze x m 0 3. Mnmze x m 0 (b) energy method y y In 1969, Hosaka[1] proposed a curve smoothng method based on the energy extremum prncple, called energy method. The basc dea of ths method s to use the cumulatve chord length cubc splne as the reconstructon curve and the total energy of the splne as the objectve functon. Its farng process s to solve the extremum problem of the objectve functon. The mechancal model of ths method s very ntutve. Assumng M 0,1,, n N 0,1, L, n, that the sequence of shape ponts before and after smoothng a curve s L and the formula for calculatng stran energy s as follows: n 1 1 E k dx N M (3) 0 In (3), Stffness coeffcent of the curve; Elastc coeffcent; k Curve curvature. () Local farng Two commonly used global farng algorthms are ntroduced. Both least squares method and energy method are sutable for the case of relatvely large number of non-farng ponts. However, when there are few non-farng ponts (such as only a lttle non-farng), the above two methods wll undoubtedly result n a large amount of computaton and speed of operaton. Slow down. Several common local smoothng algorthms are ntroduced below. (a) Pont revson method The smoothng process of the pont selecton method s: Step1: fnd out the "bad ponts" one by one; Step: modfy the "bad ponts". These two processes are dscussed below. Step1:Dscrmnaton of "bad ponts" There are two methods to dstngush "bad ponts": () Users decde "bad ponts" by themselves accordng to ther observatons, whch s called nteractve method. () Accordng to the correspondng farng crtera, a crteron for judgng "bad ponts" s establshed to accurately dentfy all "bad ponts", whch s called automatc method. Q 0,1, L, n s assgned to the set value, assumng that Q t s the curve of nterpolaton and Q, and k s the relatve curvature at the pont of type value. For automatc methods, n general, we use the followng "bad ponts" crtera: 1) The k dscontnuous type value ponts are called the 1 knd of bad ponts. ) In the sgn sequence sgn k of curvature, the pont Q of contnuous sgn change s called two knds of bad ponts even f the condton k 1k 0 and kk 1 0 hold. 3) The ponts of contnuous sgn change n the frst order dfference symbol sequence sgn k DOI: / Page of k, even f condton k 1 k 0 and k k 1 0 hold, are called three knds of bad ponts. Step: Modfcaton of "bad ponts" After dentfyng the "bad ponts", we need to modfy them. At present, the commonly used "bad pont" modfcaton methods nclude kejellander method[], node deleton and nserton method[3], roundness method and base splne method. The advantage of ths method s that the crteron s smple, local modfcaton,

5 fast calculaton speed and farng effect s very good n the case of fewer "bad ponts". The dsadvantage of ths method s that when there are many "bad ponts" n successon, the crteron of "bad ponts" depends on the curvature of adjacent ponts to calculate, so the crteron wll appear certan. The degree of msjudgment s not very good. (b) Curvature method The objectve functon of energy smoothng s to calculate the square and ntegral of curvature. The objectve of smoothng s to reduce the stran energy of the curve as much as possble, that s, to reduce the curvature of the curve, so that the curve area s smooth and the shape change s large. Chen Lang[4] developed a curve smoothng method based on the curvature method of wavelet decomposton, extracted the part wth lower frequency after wavelet decomposton as a new curvature map, and fnally reconstructed the smoothed product contour curve. The basc dea of these methods s to calculate the curvature map of the curve, then smooth the curvature map of the curve, not drectly smooth the curve. Then, accordng to the curvature map after smoothng and combnng wth the orgnal value ponts, the orgnal curve can be nversely calculated, so that the farng curve of the orgnal curve can be obtaned. V. Concluson The development of curve and surface modelng technology provdes powerful support for modern manufacturng ndustry. It extends from tradtonal curve nterpolaton, approxmaton and fttng to varous smoothng technques, and presents the phenomenon of cross-fuson of mult-dscplnary and mult-feld. The ndustry has also put new technology and theory nto ndustry applcaton very quckly, whch shows the mportance of curve and surface modelng technology for modern ndustry. Throughout the development hstory of curve and surface modelng technology, fndng a better curve farng technology s the key to promote the whole CAGD research. References [1]. Bohem, W, Farn, G and Kahman, J. A survey of curve and surface methods n CAGD, Computer Aded Geometrc Desgn, 1984, 1(1): []. Ferguson J. Multvarable curve nterpolaton. Report D-504, The Boeng Co. Seattle, Washngton, [3]. Ferguson J. Multvarable Curve Interpolaton[J]. Journal of the Acm, 1964, 11(): 1-8. [4]. Coons S A. Surface for Computer-aded Desgn of Space Fgures[J] [5]. Coons S A. Surfaces for Computer-Aded Desgn of Space Forms[M]. Massachusetts Insttute of Technology, [6]. Bezer. P. E. Numercal Control: Mathematcs and Applcatons. New York: John Wley, 197. [7]. Boor C D. On calculatng wth B-splnes[J]. Journal of Approxmaton Theory, 197, 6(1): [8]. Cox M G. The numercal evaluaton of B-splnes[J], Journal of the Insttute of Mathematcs and Its Applcatons, 197, 10(): [9]. Gordon W J, Resenfeld R F. B-splne Curves and Surfaces[J]. Computer Aded Geometrc Desgn, 1974, 3(91): [10]. Gordon W J, Resenfeld R F. Bernsten-Bézer Methods for the Computer-Aded Desgn of Free-Form Curves and Surfaces[J]. Journal of the Acm, 1974, 1(): [11]. Versprlle K J. Computer-aded desgn applcatons of the ratonal b-splne approxmaton form[m]. Syracuse Unversty, [1]. Pegl L, Tller W. Curve and surface constructons usng ratonal B-splnes[J]. Computer Aded Desgn, 1987, 19(9): [13]. Pegl L. Interactve Data Interpolaton by Ratonal Bezer Curves[J]. IEEE Computer Graphcs & Applcatons, 1987, 7(4): [14]. Tller W. Ratonal B-Splnes for Curve and Surface Representaton[J]. Computer Graphcs & Applcatons IEEE, 1983, 3(6): [15]. Tller W. Knot-removal algorthms for NURBS curves and surfaces[j]. Computer Aded Desgn, 199, 4(8): [16]. L Pepe. Research on Fttng Parameterzaton and Shape optmzaton problems of curve modelng[d]. Shandong Unversty, 01. [17]. Farn G, Ren G, Sapds N, et al. Farng cubc B-splne curves[j]. Computer Aded Geometrc Desgn, 1987, 4(1): [18]. Su Buqng, Lu Dngyuan. Computatonal Geometry[M]. Scence and Technology Press, 198. [19]. Sh Fazhong. CAGD & NURBS[M]. Hgher Educaton Press, 001. [0]. Lu Baoja, Xu Zongjun, Wang We, et al.research on the Farness of Curve and Surface[J]. Machnery Electroncs, 001(5): [1]. Hosaka M. Theory of curve and surface synthess and ther smooth fttng[j]. Ipsj Magazne, 1969, 10: []. Kjellander J A P. Smoothng of cubc parametrc splnes[j]. Computer Aded Desgn, 1983, 15(3): [3]. Sapds N, Farn G. Automatc farng algorthm for B-splne curves[j]. Computer Aded Desgn, 1990, (): [4]. Chen Lang. Research on farng of curves and surfaces wth target curvatures[d]. Hebe Unversty of Technology, 014. WuYze.. IOSR Journal of Mechancal and Cvl Engneerng (IOSR-JMCE), vol. 15, no. 6, 018, pp DOI: / Page

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