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Geometric modeling in product design activities
The most common methods of representation of geometric entities (curves and sur-faces) in geometric modeling are parametric, implicit and explicit methods. The explicit methods (Foley, van Dam, Feiner, & Hughes, 1992) that express, for example, y = f(x) (and z = g(x,y) for surfaces) are quite limiting. For instance, it is impossible to get multiple values of y for a given x, hence circles and ellipses must be represented with multiple curve parts. Fur-thermore, such representations are not rotationally invariant and describing curves with vertical tangent is difficult, because a slope of infinity is difficult to be represented. This leaves us with parametric and implicit methods as the two key approaches for geometric modeling tasks. An implicit representation (Hoffmann, 1993) of a surface is in the form F(p) = f(x, y, z) = 0 (for instance, x2 + y2 – r2 = 0 for a circle or x2 + y2 + z2 – r2 = 0 for a sphere), where p is a point on the surface implicitly defined by the expression F(p). If the relation F(p) = 0 holds for p, it means that the point p is on the surface. This representation provides the possibility for testing whether a given point p is on the surface or not. However, this repre-sentation does not provide a direct way for systematically generating points on the surfaces. Usually, an implicit representation is constructed so that it has the property that if for given x:
F(p) > 0, then the x is above the curve (surface)
F(p) = 0, then the x is exactly on the curve (surface)
F(p) < 0, then the x is below the curve (surface)
Thus, an implicit representation allows for a quick and easy inside-outside (or above – below) test. Additionally, the implicit methods provide mathematical tractability very suita-ble in computer animation and for modeling operations such as blending, sweeping, inter-section, Boolean operations. The implicit methods can be used for modeling any arbitrary closed surfaces as a single patch, but for such created surfaces are less interactive because of the possible self-interactions and other singularities (Bajaj, Chen, & Xu, 1994; Bajaj, Chen, & Xu, 1994). Such limitations make implicit surfaces not suitable for product design, where the fine control of the shape and its conversion to other representation for simulation pur-poses are necessary. This motivates the large adoption of parametric methods (Hoffmann, Geometric and Solid Modeling: An Introduction, 1989), in commercial CAD (Computer-Aided Design) systems. Other properties that motivate the parametric methods of representation to be widely used in CAD systems are their coordinate system independence, vector-valued function, ease of handling vertical slopes, and (the most important) efficient generation of points on a curves (surfaces) which is crucial for shape representation in product design. In the parametric method, each of the coordinates of a point on the curve (surface) is repre-sented separately as an explicit function of an (two) independent parameter(s).Thus, para-metric curves and surfaces are defined as follows:
P(u) = (x(u), y(u)) for curves or P u, = u, , u, z u, for surfa es ith a ≤ u, ≤
For e a ple for a sphere u = φ a d = θ
φ, θ = r si φ os θ ,
φ, θ = -r os φ ,
z φ, θ = -r si φ si θ , ≤ θ ≤ π a d ≤ φ ≤ π
Where, P(u) (or P(u,v)) is a vector-valued function of the independent variable u (or u,v). Although the interval [a, b] is arbitrary, it is usually normalized to [0, 1]. Unlike the implicit representation, the parametric representation allows us to directly generate points on the curve (surfaces). All that is required is to choose the values of the parameters u and v and then P(u, v) = F(u, v). If it is done in a systematic way over the range of possible u and v val-ues, it is possible to generate a set of points sampling the entire curve (surface). The main shortcoming of the parametric curve (surface) is the inability of positioning an arbitrary point in space in terms of testing whether the point is on the curve (surface). Piegl and Tiller, in the NURBS book, summarize and list the advantages and disadvantages of the parametric representation method (Piegl & Tiller, The NURBS Book, 1997) and the most significant are listed in the following:
1. The parametric method is easy to extend from two-dimensional space into three-dimensional space by simply adding a z coordinate.
2. The parametric method is very practical to represent bounding curve segment (or sur-face patches)
3. Parametric curves (surfaces) have ordering of the sequence that is gained from the or-dering of the parameter u in the knot vector (u and v for surfaces). Such sequence or-dering helps generating ordered sequences of points of a curve.
4. From computational point of view, the parametric method is much more convenient due to the numerically stable algorithms used to represent the geometry in a comput-er.
5. Computing a point on a curve or surfaces is an easy task in the parametric method but testing whether the point is on a curve or surface is very difficult in the parametric method.
6. The parametric method sometimes has to deal with parametric anomalies which are related more to the computation algorithm imperfection. For instance, the poles of a sphere are same as the other points on the sphere, but are algorithmically difficult.
Since both parametric and implicit forms are complement to each other in different application, it is very practical to transform one form to the other. The possibility of having mutual conversion helps combining their advantages in order to obtain better geometric representation of shapes. The conversion from parametric to implicit form is known as im-plicitization, whereas the inverse conversion is known as parametrization (Sederberg, Anderson, & Goldman, 1984).
Parametric representations
The parametric representation of the geometric entities has been adopted by almost all CAD systems enabling all analytic shapes to be represented, as well as more complex free-form entities. The Bézier, B-Splines and NURBS representations are the most used in free-form curve and surface definition (Piegl & Tiller, The NURBS Book, 1997), and they are presented in the following sections.
Bézier, B-Spline and NURBS curves
The basics of the Bézier curves have been introduced in the theoretical mathematics long time before their implementation in computer graphics by the French mathematician Charle Hermite and the Russian mathematician Sergei Bernstein. Later, the mathematical formulation of the Bézier curves has been proposed by the French engineer Pierre Bézier at Renault Car Company. There are many ways of interpreting the Bézier curves depending on which aspect is considered, the engineers prefer to interpret the Bézier curves in terms of the center of mass of a set of point masses. For instance, we can consider four masses (m0, m1, m2, and m3) positioned at four points P0, P1, P2, P3 (Figure 2.1). The position (P) of the center of mass is given by the following equation: The position (P) of the center of mass remains unchanged if the masses at each point are constant. Supposing that instead of keeping the masses constant, each mass varies aco di g to so e fu tio of a gi e pa a ete u . Fo e a ple, assu i g that the asses at each points vary according following functions: m0 = (1 – u)3, m1 = 3u(1 – u)2, m2 = 3u2(1 – u), m3 = u3. The alues of these asses as a fu tio of u are shown in this graph (Figure 2.2): If the alue u varies between 0 and 1 then the position of the center of mass (P) changes continuously sweeping out a curve. This curve is called a cubic Bézier curve (Figure 2.3). These masses (mi) function are called blending functions whereas their positions (Pi) are known as control points of the Bézier curves. The lines connecting two neighboring control points create a figure know as control polygon. In the case for Bézier curves, the blending functions are also known as Bernstein polynomials.
B-Spline curves
The limitations of the Bézier curves are overcome by formulating of the B-spline curves which are considered as a generalization of the Bézier curves. In fact, B-spline curves have more desired properties than Bézier curves gained by undertaking the Bézier curves proper-ties and overcoming the two main limitations of the Bézier curves. Namely, replacing a Bé-zier curve of higher degree with a series of many Bézier curves (B-spline) of lower degree reduces the overall degree of the curve and improves the property of local modification. A B-spline curve of degree p with n control points consists of a series of n – p Bézier curve seg-ments. For instance, a cubic curve (degree p = 3) with n = 7 control points has 4 Bézier curve segments (Figure 2.5) and these segments all have C2 continuity at the join points. The con-tinuity degree is in terms of curve degree and repetition of knots in the knot vector. In gen-eral, it can be stated that the B-spline curve requires more computation, but is far more flex-ible and pleasant to work with, which is the reason why it has become part of almost every serious graphics development environment.
Before presenting the B-spline curve, first of all, it is necessary to define both the knot vector and the B-spline basis function. The joint points between these segments are called knots and they play a fundamental role in the understanding of this kind of curves. Consider-ing U = [u0, …, um] to be a non-decreasing sequence of real numbers, that is ui ≤ ui+1, i = , …, m; where ui are called knots, and U is referred to as the knot vector. The knot vector is used to specify values in the evaluation interval where the curve changes the spline segment. By spacing the m intervals of the knot vector with equal distance, we obtain a uniform B-spline curve (for instance, U = {0,0,0,0,2,4,6,8,8,8,8}), while an uneven spacing (for instance, U = {0,0,0,0,1,5,6,8,8,8,8}) yields a non-uniform B-spline curve. Repeating a knot in the knots vec-tor (i.e. increasing the multiplicity) reduces the continuity of the curve at that knot. Thus, repeating the knots p times at the ends of the knots vector will make the B-spline to pass through the endpoints of the control polygon (for instance, U = {0,0,0,0,2,4,6,8,8,8,8}, the 0 is repeated three times same as the 8) .
When constructing a uniform B-spine curve, the basic functions translate each other, yielding a very simple basis function. The Cox de Boor algorithm can be found below, which can recursively computes any uniform (Figure 2.6) or non-uniform (Figure 2.7) B-spline basis function of n degree.
Geometric modeling strategies
Geometric Modeling consists in defining realistic visualization of shapes in a computer graphic environment. In the background, the geometric modeling deals with mathematical methods for shape representation. The concept of geometric modeling has been initially introduced in the mid-1970s with the development of computer systems. When designing complex product shapes that are not fully known in advance or do not exist yet, the usual geometric modeling procedure is interactive and involves the following steps (Patrikalakis, 2003):
• Construction of the shape based on design requirements
• Evaluating the overall shape of the object
• Improving the shape by modifying it until a satisfactory design is achieved
The aim of the Geometric modeling is to provide a complete, flexible and real repre-sentation of the object that is even not fully known, so that the shape of the object can be:
Realistic visualized (rendering)
Flexible for manipulation (modification)
Easily convert to a model that can be analyzed computationally
Successfully manufactured and tested
Surface Modeling in product design
Before computer-aided surface modeling existed, automobile profiles had to be de-signed by making a clay model of car bodywork and, by trial and error methods, this model was shaped until aesthetically and functionally satisfactory. The geometry of the car body panels was then determined by physical measurement of the clay and stored in form of 2D technical drawings and documentations. The development of computer hardware and graphic introduces the possibility for developing computer-aided geometric modeling sys-tems. The development of the geometric modeling provides a fast and flexible platform for storing the geometry of a physical model that can be later used for visualization or manipu-lation. The basic geometri odeli g st ategies i luded i toda s o pute -aided design systems are: wireframe, surfaces and solid modeling. Wireframe modeling represents ob-jects by edge curves and vertices on the surface of the object. The wireframe model repre-sents the edges of the object in Euclidian 3D space, which is much better than projection of the edges onto a plane. The 3D wireframe model are nonetheless limited in that they cannot provide an unambiguous representation of the object and do not contain any information about the interior of the object (spatial addressability). Surface modeling defines objects with greater mathematical integrity as it models the surfaces to give more definitive spatial boundaries to the design. Unlike the wireframe modeling, the surface modeling enables vis-ualizing more complex surfaces without producing a physical object (e.g. industry). The ben-efit of surface modeling using a computer is the possibility of having both the aesthetics and 3D geometry of the surfaces shape defined in one process. The surface models are types of 3D geometric models of a shape with no thickness. Clear differentiation between surface models and tick models has to be made because the latter have mass property, possibility of collision detection, and possibility for generation of finite element mesh. Solid modeling is the most advanced geometric modeling method for modeling in three dimensions. The com-plete geometric data representation of an object is ability to classify points in space relative to the object in terms if the point is inside, outside or on the object. The solid models are capable of handling spatial addressability as well as verifying that the model is well formed. The latter means that these solid models cannot verify whether two objects occupy the same space. Since the application domain of this work is the industrial design of products, the surface models are considered as a medium for transferring the aesthetic properties and the emotional affection to the customers. The surface models can be created using two dif-ferent groups of techniques divided in: free-form and subdivision surfaces, on the on hand, and B-Rep and declarative modeling, on the other hand. Therefore, these surface modeling techniques are listed in the following:
Free-form surface
Free-form surfaces are widely used in all engineering design domains where the shape of the final product is more important from costumer point of view in terms of aesthetics. Therefore, the free-form surfaces modeling is also called the art of the engineering free-form surfaces. In general, there are two main methods for creating free-form surfaces: Lofting and Sweeping. Lofting represents an exact interpolation of a surface by using set of curves. Sweeping operation consists of creating a free-form surface by sweeping a profile curve along a trajectory curve in space. The most common types of spline that are using in design-ing free-form surface are the NURBS (Section 2.2.2). This is very important in cases where there is intersection of splines and analytic surfaces. The free-form surfaces are defined by control polygon and their modification depends more on the way how the free-form surfac-es is created. In case the surface is created using either loft or sweep, the modification of the surfaces is possible by modifying the profile curves (loft and sweep), or the trajectory curves (sweep). An alternative way of modifying of all free-form surfaces, regardless the way of creation, is possible by moving the control points (Figure 2.11). Such modification of free-form surfaces is very tedious, less intuitive and limits the creativity of the designers. The modification of free-form shapes, performed by manipulation low-level geometric entities (points and curves), does not provide certainty that the final shape will satisfy the aesthetic requirements. Additionally, this character of the modification of free-form shapes discour-ages the non-designers to use them in designing their own products.
Table of contents :
1 Introduction
2. Geometric modeling in product design activities
2.1 Geometric representation methods
2.2 Parametric representations
2.2.1 Bézier, B-Spline and NURBS curves
2.2.1.1 B-Spline curves
2.2.1.2 B-Spline curves
2.2.1.3 NURBS curves
2.2.2 Bézier, B-Spline and NURBS surfaces
2.2.2.1 Bézier surfaces
2.2.2.2 B-spline surfaces
2.2.2.3 NURBS surfaces
2.3 Geometric modeling strategies
2.3.1 Surface Modeling in product design
2.3.1.1 Free-form surface
2.3.1.2 Subdivision surfaces
2.3.1.3 Boundary Representation (B-Rep)
2.3.2 Procedural design process
2.3.3 Needs for intuitive modification of geometric models
2.4 Synthesis and Conclusion
3 Aesthetic-oriented free-form shape description and design
3.1 Toward high-level modification of geometric models
3.1.1 Feature-based approaches
3.1.2 Declarative design process
3.1.2.1 Description tools
3.1.2.2 Generation techniques
3.1.2.3 Understanding tools
3.1.3 Target-driven design
3.1.4 Aesthetic-oriented design and modification of free-form shapes
3.1.4.1 Mapping of aesthetic properties to 3D free-form shapes
3.1.4.2 Defining Aesthetic Curves and Surfaces
3.2 Aesthetic properties of curves
3.2.1 Straightness
3.2.2 Acceleration
3.2.3 Convexity/Concavity
3.2.4 Other aesthetic properties of curves
3.2.4.1 Hollowness
3.2.4.2 Crown
3.2.4.3 S-Shaped curves
3.2.4.4 Tension
3.2.4.5 Lead-in
3.2.4.6 Sharpness/softness
3.2.5 Synthesis
3.3 Aesthetic properties of surfaces
3.4 Conclusion
4 Machine Learning Techniques
4.1 Data Mining and Knowledge Discovery
4.2 Categories of Machine Learning Techniques
4.3 Use of WEKA to identify classification rules and meaningful attributes
4.4 Different classification techniques
4.4.1 Single label classification
4.4.1.1 C4.5 decision trees or J48
4.4.1.2 IBk or k – Nearest Neighbors (k-NN) classification
4.4.1.3 SMO or Support Vector Machine (SVM)
4.4.1.4 NaiveBayes or Naïve Bayes (NB)
4.4.1.5 RIPPER or Decision Rules (JRip)
4.4.1.6 Training a classification model (classifier)
4.4.1.7 Classification efficiency analysis
4.4.1.8 Relevant attribute selection
4.4.2 Multi-label classification
4.4.3 Multi-dimensional classification
4.5 Applications in various domains
4.6 Conclusion and Synthesis
5 Classification framework specification and its validation on curves
5.1 Overall framework
5.2 Setting up of the framework
5.2.1 Space of shapes
5.2.2 Dataset of curves
5.3 Attributes
5.4 Classification
5.5 Considered learning methods
5.6 Experimentations
5.6.1 Modelig of the staightesss ules idetifiatio pole
5.6.2 Classification using dimensional attributes
5.6.3 Classification using dimensionless attributes
5.6.4 Relevant Attribute selection
5.7 Conclusion
6 Classification of surface shapes
6.1 Challenges for surfaces (versus curves)
6.2 Framework application
6.3 Generation of the instances data set
6.3.1 Diversity of shapes explored
6.3.2 Definition of the deformation paths and of the morphing process to generate shape surfaces
6.3.3 Definition of the surrounding surfaces
6.3.4 Generation of the initial Dataset of shapes
6.4 Definition of surface parameters using basic geometric quantities – Attributes
6.4.1 Geometric quantities and surface parameters
6.5 Classification of the surfaces by carrying out interviews
6.6 Experimentations
6.6.1 Organization of the Initial Data Set (IDS)
6.6.2 Pre-processing of the acquired data from the classification
6.7 Results and discussion
6.7.1 Comparison of the learning capability of different learning algorithms
6.7.2 Perception of flatness of every participant in the interviews
6.7.3 Influence of the surrounding (context) to the perception of flatness
6.7.4 Influence of different surrounding (objects) to the perception of flatness
6.7.5 Choosing the most relevant surface parameters
6.8 Conclusion
7 Conclusion and Perspectives
8 References
