外文翻譯---在逆向工程中對擬合曲線的數(shù)據(jù)點(diǎn)云的預(yù)處理

1、<p><b>　　中文3040字</b></p><p><b>　　附 錄</b></p><p>　　The Pre-Processing of Data Points for Curve Fitting in Reverse Engineering</p><p>　　Reverse engineeri

2、ng has become an important tool for CAD model construction from the data points, measured by a coordi-nate measuring machine (CMM), of an existing part. A major problem in reverse engineering is that the measured points

3、having an irregular format and unequal distribution are dif?-cult to ?t into a B-spline curve or surface. The paper presents a method for pre-processing data points for curve ?tting in reverse engineering. The proposed m

4、ethod has been developed to process the mea</p><p>　　1. Introduction</p><p>　　With the progress in the development of computer hardware and software technology, the concept of computer-aided tec

5、h-nology for product development has become more widely accepted by industry. The gap between design and manufactur-ing is now being gradually narrowed through the development of new CAD technology. In a normal automated

6、 manufacturing environment, the operation sequence usually starts from product design via geometric models created in CAD systems, and ends with the generation of m</p><p>　　1）. Where a clay model, for examp

7、le, in designing automobile body panels, is made by the designer or artist based on conceptual sketches of what the panel should look like.</p><p>　　2）. Where a sample exists without the original drawing or

8、documentation de?nition.</p><p>　　3）. Where the CAD model representing the part has to be revised owing to design change during manufacturing.</p><p>　　In all of these situations, the physical m

9、odel or sample must be reverse engineered to create or re?ne the CAD model. In contrast to this conventional manufacturing sequence,everse engineering typically starts with measuring an existing physical object so that a

10、 CAD model can be deduced in order to exploit the advantages of CAD technologies. The process of reverse engineering can usually be subdivided into three stages, i.e. data capture, data segmentation and CAD modeling and/

11、or updating. A phy</p><p>　　In practical measuring cases, however, there are many situ-ations where the geometric information of a physical prototype or sample cannot be measured completely and accurately to

12、 reconstruct a good CAD model. Some data points of the measured surface may be irregular, have measurement errors,or cannot be acquired. As shown in Fig. 1, the main surface of measured object may have features such as h

13、oles, islands,or roughness caused by manufacturing inaccuracy, consequently</p><p>　　the CMM probe cannot capture the complete set of data points to reconstruct the entire surface.</p><p>　　Fig.

14、 1. The general problems in a practical measuring case.</p><p>　　Measurement of an existing object surface in reverse engin-eering can be achieved by using either contact probing or non-contact sensing probi

15、ng techniques. Whatever technique is applied, there are many practical problems with acquiring data points, for examples, noise, and incomplete data. Without extensive processing to adjust the data points, these problems

16、 will cause the CAD model to be reconstructed with an unde-sired shape. In order to rebuild the CAD model correctly and satisfactorily, th</p><p>　　2. The Theory of B-spline</p><p>　　Most of the

17、 surface-based CAD systems express shape required for modelling by parametric equations, such as in Be ´zier or B-spline forms. The most used is the B-spline form B-splines are the standard for representing freeform

18、 curves and surfaces in current commercial CAD systems. B-spline curve and Be ´zier curves have many advantages in common Control points in?uence the curve shape in a predictable natural way, making them good candi

19、dates for use in an interactive environment. Both types of </p><p>　　A B-spline curve is a set of basis functions which combines the effects of n + 1 control points. A parametric B-spline curve is given by&l

20、t;/p><p>　　For B-spline curves, the degree of these polynomials is controlled by a parameter k and is usually independent of the number of control points, and the B-spline basis functions are de?ned by the foll

21、owing expression:</p><p>　　Where k controls the degree (k?1) of the resulting poly-nomials in u and thus also controls the continuity of the curve.A B-spline surface is de?ned in a similar way to a tensor pr

22、oduct in a B-spline curve. It is also possible to de?ne a B-spline surface having different degrees in the u- and v-direc-tions:</p><p>　　3. Curve Fitting</p><p>　　Given a set of data points mea

23、sured from existing object,curve ?tting is required to pass through the data points. The least-squares ?tting technique is the most used algorithm which aims at approximating, based on an iterative method, a set of data

24、points to form a B-spline .Given a set of data points Qk, k = 0,1,2,. . .,n, that lie on an unknown curve P for certain parameter values uk,k = 0,1,2,. . .,n; it is necessary to determine an exact interp-olation or best

25、?tting curve, P.</p><p>　　To solve this problem, the parameter values (uk) for each of the data points must be assumed. The knot vector and the degree of the curve are also determined. The degree in practica

26、l applications is generally 3 (order = 4). The parameter values can be determined by the chord length method:</p><p>　　Given the parameter values, a knot vector that re?ects the distribution of these paramet

27、ers has the following form:</p><p>　　It can be proved that the coef?cient matrix is totally positive and banded with a bandwidth of less than p, therefore, the linear system can be solved safely by Gaussian

28、elimination without pivoting.</p><p>　　4. The Requirement for Fitting a Set ofData into a B-Spline Curve</p><p>　　In order to produce a B-spline curve with a “good shape”,some characteristics ar

29、e required to ?t the data point set into a curve presented in B-spline form. First, the data points must be in a well-ordered sequence. When applying the program to ?t a set of data points into a B-spline curve, the data

30、 points must be read one by one in a speci?ed order. If the data points are not in order, this will cause an undesired twist or an out-of-control shape of the B-spline curve.Secondly, an even dispersi</p><p>

31、;　　Having the data points equally distributed is important for improving the result of parameterisation for ?tting a B-spline curve. As the mathematical presentation shows in Eq. (9), the control points matrix [P] is det

32、ermined by the basis functions [N] and data points [Q], where the basis functions [N] are determined by the parameters ui which are correspond to the distribution of the data points. If the data points are distributed un

33、equally, the control points will also be distributed unequally </p><p>　　of a physical sample often has some features such as holes,islands, and radius ?llets, which prevent the CMM probe from capturing data

34、 points with equal distribution. If a curve is rebuilt by ?tting data points with an unequal distribution, as shown in Fig. 2, the generated curve may differ from the real shape of the measured object. Figure 3 illustrat

35、es that a</p><p>　　smoother and more accurate reconstruction may be obtained by ?tting an equally spaced set of data points.</p><p>　　5. The Pre-Processing of Data Points</p><p>　　T

36、o achieve the requirements for ?tting a set of data points into a B-spline curve as mentioned above, it is very important and necessary that the data points must be pre-processed before curve ?tting. In the following des

37、cription, a useful and effective method for pre-processing the data points for curve ?tting is presented. The concept of this method is to regress a set of measuring data points into a non-parametric equation in implicit

38、 or explicit form, and this equation must also satisfy the </p><p>　　illustrates an overview of the procedure to pre-process the data points for reverse engineering.</p><p>　　Data point ?ltering

39、 is the ?rst step in displacing the unwanted points and the noisy points. The original data points measured from a physical prototype or an existing sample by a CMM are in discrete format. When the measured points are pl

40、otted in a diagram, the noisy points which obviously deviate from the original curve can be selected and removed by a visual search by the designer for extensive processing. In addition,the distinct discontinuous points

41、which apparently relate to a sharp change</p><p>　　Many approaches have been developed for generating a CAD model from measured points in reverse engineering. A complex model is usually constructed by subdiv

42、iding the com-plete model into individual simple surfaces [8,9]. Each of the individual surfaces de?nes a single feature in a CAD system and a complete CAD model is obtained by further trimming,</p><p>　　ble

43、nding and ?lleting, or using other surface-processing tools.When the designer is given a set of unorganised data points measured from an existing object, data point segmentation is required to reconstruct a complete mode

44、l by de?ning individual simple surfaces. Therefore, curvature analysis for the data points is used for subdividing the data points into individual groups.</p><p>　　In order to extract the pro?le curves for C

45、AD model recon-struction, in this step, data points are divided into different groups depending upon the result of curvature calculation and analysis of the data points. For each 2D curve, y = f(x), the curvature is de?n

46、ed as:</p><p>　　If the data is expressed in discrete form, for any three consecutive points in the same plane(X1,Y1) ·(X2,Y2) · (X3,Y3), the three points form a circle andthe centre (X0,Y0) can be

47、calculated as (see Fig. 5):</p><p>　　Figure 6 illustrates an example in which the curvatures of a plane curve consisting of a data point set are calculated using the previous method. The curvature of the cur

48、ve determined by the data point set changes from 0 to 0.0333, as shown in Fig. 7. This indicates that there is a ?llet feature with a radius 30 in the data points set. Thus, these points can be isolated from the original

49、 data points, and form a single feature. By curvature analysis, the total array of data points is divided into</p><p>　　After segmentation, individual groups of data points are separately regressed into expl

50、icit non-parametric equations, and then the data points can be regenerated from the regression equation in a well-ordered sequence, with appropriate spacing and an equal distribution so that better ?tting can be achieved

51、.The format of the new data point set is valid for ?tting into a single simple B-spline curve without inner constraints, which can be applied for further editing and modifying, such as trimming</p><p>　　a co

52、mplete CAD model conforming to the design intent.</p><p>　　Additionally, some regression errors are introduced by the regression operation between the measured points and the regression equation. In the foll

53、owing example, the order of the regression equation is discussed, because it bears a close relationship to the regression errors. Given a set of existing data points, the set is regressed using a different order of the&l

54、t;/p><p>　　regression (order = 2,3,4,5). Figure 8 illustrates the relationship between the order of the regression equations and the regressed calculated by the root-mean-square (r.m.s.) method. This</p>

55、<p>　　?gure shows that increasing the equation order causes a decrease of the r.m.s. error. However, in most cases, when the 5th-order of the regression equation is used, the coef?cient of the 5th-order item become

56、s zero. i.e. the r.m.s. error of the 4th-order equation is equal to the 5th-order equation. This means that the designer only has to regress the data points into a 4th-order equation. In practice, a 4th-order equation ha

57、s already satis?ed the demand for curvature continuity in CAD model cons</p><p>　　6. Conclusion</p><p>　　Geometric modelling is a technology that is already used extensively in industrial applic

58、ations for developing new pro-ducts. Reverse engineering has become an important tool for CAD model construction for an existing part from the measur-ing data. A major dif?culty in reverse engineering techniques is to ?t

59、 the irregular data points of an unequal distribution into a B-spline curve. The procedure of the pre-processing of data points for curve ?tting in reverse engineering is described in this pa</p><p>　　and is

60、 an effective tool for integrating with current commercial CAD systems for reconstructing the geometric models of physi-cal parts.</p><p>　　A broader interpretation of the term “reverse engineering”might per

61、haps involve deducing the intent of the original designer to some degree. An ideal system of reverse engineer-ing would be able to not only construct a complete geometric model of the source object but also catch the ini

62、tial design intent. By applying the method proposed above, designers may regroup the data points in order to produce the individual</p><p>　　feature curves for reconstructing a complete CAD model of the sour

63、ce object to achieve the original design intent.</p><p>　　在逆向工程中對擬合曲線的數(shù)據(jù)點(diǎn)云的預(yù)處理</p><p>　　逆向工程已經(jīng)成為一種從現(xiàn)存物體通過CMM測量的數(shù)據(jù)點(diǎn)重建CAD模型的重要工具.在逆向工程中首要的問題是:測量到的點(diǎn)具有不規(guī)律形式和不對等分布很難用B-spline曲線擬合。這篇論文中介紹了一種在逆向工程中用預(yù)先處

64、理數(shù)據(jù)點(diǎn)來擬合曲線的方法。適合B-spline形式之前來處理先前測量得到的數(shù)據(jù)點(diǎn)的方法已經(jīng)得到了發(fā)展。通過這種方法產(chǎn)生的新的數(shù)據(jù)點(diǎn)形式，適合建立光滑精確B-spline曲線的要求。這種方法的整個的步驟包括：切片，弧度分析，分割，回歸，和再生。在逆向工程中這種方法被實(shí)施和用于實(shí)踐應(yīng)用。重建的結(jié)果證實(shí)了此方法與目前流行的商業(yè)CAD系統(tǒng)的結(jié)合能力。</p><p><b>　　1．介紹</b>&l

65、t;/p><p>　　隨著計算機(jī)硬件的軟件技術(shù)的發(fā)展，對促進(jìn)產(chǎn)品發(fā)展的計算機(jī)輔助技術(shù)觀念在工業(yè)領(lǐng)域已被廣泛地接受通過新的CAD技術(shù)的發(fā)展,設(shè)計和制造之間的間隙已逐漸變得越來越密切。在正常的自動化制造環(huán)境下操作順序經(jīng)常是通過用CAD系統(tǒng)創(chuàng)建的幾何模型的產(chǎn)品設(shè)計開始，在幾何模型的基礎(chǔ)上，產(chǎn)生機(jī)器制造指令將原材料轉(zhuǎn)化成最終產(chǎn)品然后結(jié)束。由于意識到現(xiàn)代計算機(jī)輔助技術(shù)在產(chǎn)品發(fā)展和制造中的優(yōu)勢，因此在CAD系統(tǒng)著重要求創(chuàng)建物體的

66、幾何模型。然而，在創(chuàng)建CAD 模型之前，產(chǎn)品發(fā)展的物理模型和樣本先被產(chǎn)生出來。</p><p>　　(1)．例如，在設(shè)計汽車主體控制面板時，設(shè)計者和藝術(shù)家關(guān)于控制板的構(gòu)想到底是在什么樣的基礎(chǔ)上制造黏土模型。</p><p>　　(2).沒有最初的草圖，確切的記錄模型在哪里？</p><p>　　(3).在制造中由于設(shè)計的改變，CAD模型不得不重新修改的部分哪里？&l

67、t;/p><p>　　在所有這些情形中。物理模型或樣本的建立是為了創(chuàng)建和建立CAD模型。與這些常規(guī)的制造順序相反，典型的逆向工程從測量現(xiàn)存的物理實(shí)體開始，這樣推斷出來的CAD模型可以更好的利用CAD技術(shù)的優(yōu)勢。逆向工程經(jīng)常可以細(xì)分為3個階段：電子數(shù)據(jù)獲取，數(shù)據(jù)分割，和用CAD模型構(gòu)建一個物理模型。樣本起先用CMM或激光掃描儀測量以得到以三維坐標(biāo)形式存在的幾何圖案的信息。然后，為了更進(jìn)一步的處理,測量結(jié)果被分割成拓?fù)錉?/p>

68、。就重建模型來說，每個小區(qū)域就代表一個簡單的可以用數(shù)學(xué)方面知識描繪其簡單外表的幾何圖案特征。CAD 模型重建區(qū)域的表面是把這些表面連接成完整的可以描述被測量部分或樣本的模型。</p><p>　　然而，在實(shí)際測量方案中，存在物理樣本或模型的幾何圖案信息不能被完全測量和準(zhǔn)確重建一個好的CAD 模型的情況。一些表面測量的數(shù)據(jù)可能是不規(guī)律的，還有一些測量誤差或者表面是不要求的。如圖1所示，測量物體的主要表面可能有這些特

69、征：由于制造的不精確引起的凹坑，凸起，或噪聲點(diǎn)，因此，CMM探針不能獲取一套完全的數(shù)據(jù)點(diǎn)來重建整個物體的表面。</p><p>　　圖1在實(shí)際測量情況中的一般的問題</p><p>　　在逆向工程中，現(xiàn)存實(shí)體的測量，可以通過接觸式測量或非接觸式測量技術(shù)來實(shí)現(xiàn)。然而無論用什么技術(shù)，這里都有一系列獲取數(shù)據(jù)的實(shí)際問題，例如，噪聲和不完全數(shù)據(jù)。如果沒有簡單的程序去校對這些數(shù)據(jù)點(diǎn)。這些問題將引起令人

70、不期望的CAD 模型的重建問題。為了正確和滿意的重建CAD模型，這篇論文中介紹了一種先處理數(shù)據(jù)點(diǎn)去擬合曲線的有用和行之有效的方法，用這種方法，數(shù)據(jù)點(diǎn)被按指定的形式重新生成，并適合指定擬合B-spline曲線的形式，而沒有先前提到的問題。在擬合了所有曲線之后，模型的表面才可能完全和曲線結(jié)合起來。 </p><p>　　2．B-spline曲線理論</p><p>　　通過含參數(shù)的方程，絕大多

71、數(shù)外觀基礎(chǔ)上的CAD系統(tǒng)都表達(dá)了構(gòu)造模型的要求，如Bezier曲線或 B-spline曲線形式，最長用的是B-spline形式，在目前商業(yè)系統(tǒng)中，B-spline曲線是標(biāo)準(zhǔn)的代表自由曲線和外表的曲線。B-spline曲線和Bezier 曲線有許多共同的優(yōu)勢。用可預(yù)測的普通方法來移動控制點(diǎn)影響曲線形狀，使它們兩者成了構(gòu)建曲面較好的曲線形式。這兩種不同類型的曲線都具有控制點(diǎn)少，獨(dú)立的對稱軸和綜合價值。都表現(xiàn)出了凸凹性。然而，在局部的控制曲

72、線形狀這方面，可能B-spline曲線表現(xiàn)出的優(yōu)勢超過了Bezier技術(shù)。如增加控制點(diǎn)而沒有增加曲線的度數(shù)的能力?？紤]到現(xiàn)實(shí)世界中應(yīng)用的要求，在這篇論文中B-spline技術(shù)被用來代表曲線和曲面。一條B-spline曲線設(shè)定了連接n + 1個 控點(diǎn)。通過下面的列子給出了一條含參數(shù)的B-spline曲線：</p><p>　　對于B-spline曲線，這些變量參數(shù)的度數(shù)經(jīng)常通過參數(shù)K控制，它對應(yīng)控制點(diǎn)的數(shù)量。一條B

73、-spline曲線基本功能通過下面的表達(dá)式來定義：</p><p><b>　　3.擬合</b></p><p>　　如果從現(xiàn)存的數(shù)據(jù)中測量一些數(shù)據(jù)點(diǎn)，擬合曲線不許經(jīng)過數(shù)據(jù)點(diǎn)。最新的擬合技術(shù)，用接近的算法規(guī)則，在迭代方法的基礎(chǔ)上，一系列數(shù)據(jù)點(diǎn)形成了B-spline曲線。假如一系列數(shù)據(jù)點(diǎn)，在一條不知道參數(shù)值的曲線P中，K從1到N決定一個準(zhǔn)確加入位置或者是好的擬合曲線P是

74、必要的。</p><p>　　為了解決這個問題，每個數(shù)據(jù)點(diǎn)的參數(shù)值必須被假定出來。矢量的節(jié)點(diǎn)和曲線的度數(shù)也是要求的。在實(shí)際應(yīng)用中度數(shù)一般都是3，參數(shù)值的確定可以通過下面的方法：</p><p>　　如果給定參數(shù)值，反映這些參數(shù)分布的節(jié)點(diǎn)如下面的形式。</p><p>　　圖2與不平等的分布曲線擬合的數(shù)據(jù)點(diǎn)</p><p>　　它可以證明,該系

75、數(shù)矩陣的完全是正面和聯(lián)合的帶寬小于p,因此,線性系統(tǒng)可以解決無鏈安全地通過高斯消元。</p><p>　　方程（5）可以寫在一個矩陣的形式：</p><p>　　在那Q是一個（m+1）×1距陣，N是一個（m+1）×(n+1) 距陣,P是一個（n+1）×1距陣,當(dāng)m>n這個方程是超出設(shè)定，它的解釋是：</p><p>　　4．適合B

76、-Spline曲線的數(shù)據(jù)要求 </p><p>　　為了生成一條光滑準(zhǔn)確的B-Spline曲線，還要求一系列數(shù)據(jù)點(diǎn)能適合呈現(xiàn)出的B-Spline形式的曲線特征。首先，數(shù)據(jù)必須有較好的排列順序。當(dāng)應(yīng)用程序?yàn)榱耸挂幌盗袛?shù)據(jù)點(diǎn)能適合-Spline曲線，這些數(shù)據(jù)點(diǎn)必須以指定的順序讀入。如果數(shù)據(jù)點(diǎn)不是按順序的，這將引起未預(yù)期的曲線或一條失去B-Spline曲線形狀控制的曲線。其次，均勻分布數(shù)據(jù)點(diǎn)對擬合曲線來說是比較好

77、的。在實(shí)際的測量中，一些因素如機(jī)器的顫抖，系統(tǒng)中的噪音，和被測量物體表面的粗糙，這都將影響測量的結(jié)果。所有這些現(xiàn)象都將引起在經(jīng)過問題點(diǎn)的曲線的局部顫抖。因此，對于產(chǎn)生一個高質(zhì)量的B-Spline曲線，光滑有序的點(diǎn)云數(shù)據(jù)是必須的。</p><p>　　獲得均勻分布的數(shù)據(jù)點(diǎn)，可以提高擬合B-Spline曲線參數(shù)的結(jié)果。就象在方程式（9）中數(shù)學(xué)方面所展示的那樣，通過和數(shù)據(jù)點(diǎn)分布一致的參數(shù)UI決定的基本函數(shù)和數(shù)據(jù)點(diǎn)，確定

78、了控制點(diǎn)。如果數(shù)據(jù)是不均勻的，這些控點(diǎn)也會分布不均勻還將引起擬合曲線的不平滑。正如上面所提及到的，在實(shí)際案例測量中 </p><p>　　一個物體模型經(jīng)常有一些諸如空洞，內(nèi)凹和小范圍的切片，這些都將阻止CMM探針獲得均勻分布的數(shù)據(jù)點(diǎn)。如果一條曲線不是用均勻分布的數(shù)據(jù)點(diǎn)擬合重建的，就像圖2中所示，產(chǎn)生的曲線會和真實(shí)測量物體的形狀不符。圖3說明了更光滑和更準(zhǔn)確的重建可以通過一系列均勻分布的空間數(shù)據(jù)點(diǎn)獲得。</p

79、><p>　　圖3與平等的分配曲線擬合的數(shù)據(jù)點(diǎn) 圖4程序的數(shù)據(jù)點(diǎn)的預(yù)處理</p><p><b>　　5.數(shù)據(jù)點(diǎn)預(yù)處理</b></p><p>　　正如上面所述，為了達(dá)到使一系列數(shù)據(jù)點(diǎn)適合B-spline曲線的要求，在擬合曲線之前，對數(shù)據(jù)點(diǎn)進(jìn)行預(yù)處理是非常重要和必須的。在下面的描述中，將介紹有種對擬合曲線有用而且有效的的數(shù)據(jù)

80、預(yù)處理辦法，這種辦法的構(gòu)想是：用絕對的或明確的形式將一系列測量結(jié)果設(shè)為不含參數(shù)的方程式，這些方程式必須滿足曲率的連續(xù)性，對于一個飛機(jī)模型，一個明確的不含參數(shù)方程式的一般形式： </p><p>　　圖示說明，一個總的逆向工程中預(yù)處理數(shù)據(jù)點(diǎn)的程序。數(shù)據(jù)點(diǎn)的移動第一步是刪除不需要和不規(guī)則的數(shù)據(jù)點(diǎn)。通過CMM從物理模型和現(xiàn)存模型測量的原始數(shù)據(jù)點(diǎn)是離散形式的，當(dāng)這些測量的點(diǎn)用圖表示出來時，明顯偏離原始曲線的數(shù)據(jù)點(diǎn)，可通過

81、設(shè)計者的一般處理和可見的搜尋能被有選擇的剔除掉。此外，為進(jìn)一步處理清晰的不連續(xù)的在形狀上急轉(zhuǎn)變化的點(diǎn)，可以很容易的把他們分開。</p><p>　　在逆向工程中，從測量點(diǎn)中產(chǎn)生一個CAD模型已經(jīng)發(fā)展了很多種途徑。一個復(fù)雜的模型經(jīng)常要通過將完整的模型細(xì)分成單獨(dú)的簡單模型來構(gòu)建。在一個CAD系統(tǒng)中，每一個單獨(dú)的表面定義了一個簡單的特性。一個完整的的CAD模型就可以通過更進(jìn)一步的修整，融合，整合獲得，或者用其他的表面處

82、理工具。當(dāng)一個設(shè)計者把從存在的物體中測量的一系列數(shù)據(jù)進(jìn)行細(xì)分時，要求通過定義單獨(dú)的簡單表面來重新構(gòu)建一個完整的模型。 因此，數(shù)據(jù)點(diǎn)的曲率分析被用來將細(xì)分的的數(shù)據(jù)點(diǎn)歸成單獨(dú)的小類。</p><p>　　為了提煉出再建的CAD模型，在這一步中，依據(jù)曲率推算和數(shù)據(jù)點(diǎn)分析的結(jié)果，數(shù)據(jù)點(diǎn)被歸為不同的類，如一個2維作標(biāo)的曲線 </p><p><b>　　曲線被定義如下：</b>

83、</p><p>　　如果數(shù)據(jù)用離散的形式表示出來，同一架飛機(jī)中任何三個不連續(xù)的點(diǎn)，這三點(diǎn)形成一平面和一個中心</p><p>　　a = (X1 + X2) (X2 - X1) (Y3 - Y2) </p><p>　　b = (X2 + X3) (X3 - X2) (Y2 - Y1) </p><p>　　c = (Y1 - Y3) (Y

84、2 - Y1) (Y3 - Y2) </p><p>　　d = 2[(X2 - X1) (Y3 - Y2) -(X3 - X2) (Y2 - Y1)] </p><p><b>　　圖6圓角的模型</b></p><p>　　e = (Y1 + Y2) (Y2 - Y1) (X3 - X2) </p><p>　　f

85、= (Y2 + Y3) (Y3 - Y2) (X2 - X1) </p><p>　　g = (X1 - X3) (X2 - X1) (X3 - X2) </p><p>　　和, (X2, Y2)的曲率K可以定義為:</p><p>　　圖6說明了一個例子，組成數(shù)據(jù)點(diǎn)的飛機(jī)輪廓的曲度用先前方法推算，數(shù)據(jù)點(diǎn)從0到0.333之間的變化決定了曲線的曲度，就像圖7中所示。

86、這表明數(shù)據(jù)點(diǎn)中有一些半徑為30的點(diǎn)。然而，這些數(shù)據(jù)可以從原始數(shù)據(jù)中分離出來而形成一個簡單的特性。通過弧度分析，這一組數(shù)據(jù)點(diǎn)被分成了幾類。從外觀上急劇變化的原始數(shù)據(jù)的點(diǎn) 被分成了這一組組數(shù)據(jù)。在分割完以后，單獨(dú)的數(shù)據(jù)類被單獨(dú)地回歸為明確的不含參數(shù)的方程式。然而一個好的有序的，接近空間的數(shù)據(jù)點(diǎn)可以從回歸方程式中得出。</p><p>　　從而得到合適的擬合曲線。新的數(shù)據(jù)點(diǎn)對于擬合簡單的單獨(dú)的沒有內(nèi)部約束的B-spli

87、ne曲線是有效的。這些能被用于更進(jìn)一步的編輯和修改，如修飾和伸展。通過聯(lián)合單獨(dú)曲線就可以構(gòu)建出外觀，設(shè)計者不遺余力地努力實(shí)現(xiàn)一個完整的CAD模型，從而形成設(shè)計意圖。此外，通過被測量數(shù)據(jù)和回歸方程式的回歸性操作，一些回歸性的錯誤也被介紹出來，在下面的列子中，來討論回歸方程式的順序，因?yàn)樗@示出了和回歸性錯誤有密切聯(lián)系。假設(shè)一系列現(xiàn)存的數(shù)據(jù)點(diǎn)，用不同順序回歸。圖8 顯示說明了通過r.m.s.方法推算的回歸方程式和回歸性錯誤之間的關(guān)系。<

88、;/p><p>　　圖8順序和r.m.s錯誤之間的關(guān)系 </p><p>　　這數(shù)字顯示了方程式順序增加會引起r.m.s.錯誤的減少。然而，在多數(shù)實(shí)例中，當(dāng)用第5個回歸方程式的時候，第5項(xiàng)的系數(shù)變成零第 4項(xiàng)方程式的錯誤和第5項(xiàng)的錯誤是一樣的了。這就意味著設(shè)計者僅僅回歸了第4個方程式的數(shù)據(jù)點(diǎn)。在實(shí)際應(yīng)用中，第4個方程式已經(jīng)滿足了工業(yè)應(yīng)用中的CAD模型再建對曲度來連續(xù)的要求。 </p>

89、;<p><b>　　6．結(jié)論</b></p><p>　　對于開發(fā)新產(chǎn)品，構(gòu)建幾何模型已經(jīng)是一個廣泛應(yīng)用于工業(yè)的技術(shù)。逆向工程成了一個從測量到實(shí)體數(shù)據(jù)重建CAD模型的重要工具。在逆向工程技術(shù)中，一個主要的難題是：使不均勻分布的非常規(guī)的數(shù)據(jù)點(diǎn)適合B-spline曲線。在這篇論文中描述了在逆向工程中對于適合去預(yù)先處理數(shù)據(jù)點(diǎn)的過程，在擬合曲線之前處理從實(shí)體得到的數(shù)據(jù)。先前提議的方法

90、已經(jīng)得到了發(fā)展，然后，適合擬合光滑漂亮的B-spline曲線所要求的新數(shù)據(jù)被產(chǎn)生出來，這種方法的整個過程包括：切片，曲度分析，分割，回歸和再生。這種方法在逆向工程中是實(shí)際應(yīng)用的工具。也是一種連接現(xiàn)行的重建物理實(shí)體幾何模型的商用CAD系統(tǒng)的有效工具 </p><p>　　逆向工程更廣泛的解釋還可能包括：在某中程度上推斷原始設(shè)計意圖。一個逆向工程構(gòu)思體系，不僅僅是重建原始物體的完整的幾何模型，而是還要獲取原始設(shè)計意圖