TECHNIQUES FOR FREEFORM FEATURE MODELLING

Transcription

1 TECHNIQUES FOR FREEFORM FEATURE MODELLING Eelco van den Berg Faculty of Information Technology and Systems Delft University of Technology Willem F. Bronsvoort Faculty of Information Technology and Systems Delft University of Technology Joris S.M. Vergeest Faculty of Design, Engineering and Production Delft University of Technology ABSTRACT Research on regular-shaped features has been going on for a relatively long period now. However, more complicated, freeform features are demanded by product designers. Freeform shapes are mostly represented by low-level entities, such as Bézier, B-Spline and NURBS patches. Such low-level entities are not intuitive enough for designers to work with directly. Freeform features provide a high-level interface to these low-level entities. A general overview of the current state-of-the-art in freeform feature modelling is given. Then, two directions in freeform feature modelling research are looked at in more detail. First, generic feature class definition for design by freeform features is described. In comparison to regularshaped feature classes, new ways of defining geometry and specifying validity conditions have to be found. Second, freeform feature recognition by template matching is described. A semiautomatic approach is proposed to obtain features from a low-level representation of a product. KEYWORDS Feature modelling, Freeform features, Feature class definition, Design by features, Feature recognition. 1. INTRODUCTION Most of the current modelling systems are feature modelling systems. The modelling facilities of these systems are quite diverse, but a major shortcoming in nearly all of them is their limited shape domain. Because of strength, flow, aesthetic or other requirements, products often contain freeform surfaces. Current feature modelling systems, however, usually only supply regular shapes to the user, in particular prismatic and cylindrical shapes. Extending such systems with capabilities for freeform shapes is therefore much desired. Although some systems have been developed that can handle certain freeform shapes, many issues remain to be solved in order to turn such systems into full-fledged feature modelling systems. After giving a general introduction to freeform feature modelling, and a number of classifications of freeform features, this paper introduces new techniques for solving two of these issues. First, it is desirable to offer users the possibility of modelling with a set of useful, well-defined freeform features. Usually, several classes of predefined features will be available to the user, but, because of the large variety in freeform features, a good mechanism to define new classes is also required. Suitable parameters must be assigned to classes of features, in such a way that users can intuitively specify instances and modify them. A technique for defining freeform volumetric feature classes is presented. Second, it is desirable to be able to recognise freeform features in a model, e.g. to re-use a freeform feature already present in a model at another position in the model, possibly with different parameter values. A technique for doing this with freeform surface features, which is based on template matching, is presented.

2 Section 2 introduces the main concepts in feature modelling. Section 3 gives some classifications of freeform features. Section 4 describes the technique to define freeform volumetric feature classes, and Section 5 the technique for performing freeform surface feature recognition. Finally, Section 6 gives some conclusions and indicates some further developments. 2. FREEFORM FEATURE MODEL- LING Freeform feature modelling can be regarded as an extension of the prevalent way of feature modelling, in which only regular-shaped features can be used. Therefore, the main concepts of freeform feature modelling are introduced on the basis of concepts used there. A more or less accepted definition is that a feature is a representation of shape aspects of a product that are mappable to a generic shape and functionally significant for some product lifecycle phase, at least if we restrict ourselves to form feature modelling. Functional information, e.g. on the use of the shape for the end-user or on the way the shape can be manufactured, can be associated with the shape information (Shah and Mäntylä 1995). In almost all current feature modelling systems, only regular-shaped features, i.e. features having prismatic and cylindrical shapes, can be used. Freeform features are similar to regular-shaped features. They still correspond to generic shapes, the only difference being that there is more modelling freedom for the shape of the features; typically, their faces can be modelled with NURBS (Piegl and Tiller 1995). In freeform feature modelling, the general outline of a product is usually created in the initial phase of the modelling process by defining a primary feature. Later, secondary features can be added to adjust the product, while preserving the global outline of the product. Secondary freeform features are also referred to as detail features. Several types of regular-shaped features can be distinguished; protrusions, holes, slots and pockets are typical examples of frequently used ones. All properties of a feature type are specified in the corresponding feature class, which defines a template for all its instances. This always includes the generic shape of the feature and a number of parameters that characterise this shape. Similar types of freeform features can be distinguished, but now with freeform faces. In addition, however, many new types of freeform features can be defined. Although some attempts have been made (see Section 3), it has turned out to be very difficult to make a general classification of freeform features. It is therefore very important that new types of features can be introduced in a freeform feature modelling system. The definition of a new freeform feature class is, however, rather complicated: not only does the generic shape have to be modelled with, for example, NURBS, but also a set of parameters has to be chosen that makes intuitive instantiation and modification of the feature possible, and a mapping between these parameters and the low-level definition entities of the NURBS has to be established. In Section 3, an example of a simple freeform feature and its parameterisation is given. By determining values for the parameters, an instance of a feature class can be defined in a model. The instance is normally attached to other features in the model, i.e. some of its faces are coupled to faces of other features. Attachment of freeform features is again more difficult than attachment of regular-shaped features, and may require blends to realise smooth transitions. In Figure 1, two models with some freeform features are shown. Two feature parameters have been varied, namely length of the right-side arm and radius of the middle cylindrical hole. Notice that the freeform rib deforms relative to its attach faces. In advanced feature modelling systems, several properties that correspond to functional information can also be specified. In a feature class, feature validity conditions that all instances of the class should satisfy can be given. Examples are that the radius of a blind hole should be between 5 and 15 cm, and that the top face of the hole should remain open, i.e. not be covered by any other feature. In addition to feature validity conditions, there can also be model validity conditions, which specify relations on or between specific instances in a feature model. Examples of model validity conditions in regular-shaped feature modelling are that (the side faces of) two slots should be parallel and that the diameter of a hole should be half of the width of the protrusion it is attached to. Model validity conditions are specified on the instances involved. Validity conditions can be specified with constraints on feature entities, in particular faces or dimensions

3 of the features. A good set of validity conditions is indispensable for specification of features. (a) (b) Figure 1. Variations of a freeform feature model: (a) is standard model, (b) has increased length of the right-side arm, and larger radius of the middle cylindrical hole, whereas the freeform rib has been deformed accordingly The idea that validity conditions that correspond to functional information can be included in freeform feature classes, and also be set on or between specific freeform feature instances, has not yet been explored, but seems nevertheless very promising. One can easily imagine conditions that all instances from some freeform feature class have to satisfy, e.g. that the curvature of their side faces is limited, or conditions on interaction between feature instances, e.g. that the volume modified by another feature is limited. Such validity conditions can again be specified with constraints, although these may become quite complex. There are many applications of regular-shaped features. In product design, generic shapes with some function for the end-user of the product can be considered as features. In design analysis, in particular stress analysis with the finite-element method, features may represent local stiffeners or other areas relevant for the analysis. In process planning for manufacturing, volumes in a product that can be manufactured with a single or a sequence of machining operations are considered as features. Each application can have its own way of looking at a product, i.e. its own feature model of the product, with features relevant for that application. Such an application-specific feature model is called a view on the product (Bronsvoort and Jansen 1993). The potential applications of freeform features, and the corresponding views on a product, are similar to those for regular-shaped features. Independent from their application, there are basically three ways to create a feature model: design by features, feature recognition and feature conversion. In design by features, the designer specifies a feature model. He can create instances from feature classes in a library, by specifying values for the parameters, and add them to the model. In some systems, also new feature classes, so-called user-defined features (Hoffmann and Joan- Arinyo 1998), can be created by a user, after which instances from these can be added to the model. Feature instances can also be modified, by changing their parameters, or be removed from the model. The features that are specified may well have a functional meaning for the end-user of the product, but can also be manufacturing features. In the latter case, the designer specifies a model that, more or less, corresponds to the way the product will be manufactured. An important aspect of design by features is validity maintenance. The feature validity conditions specified in the feature classes of all instances in a model, and the model validity conditions specified on or between the instances, should be maintained by the system. In practice, most feature modelling systems have only a very rudimentary form of validity maintenance. Although some functional information is stored with the features in a model using constraints, the validity of models is not adequately maintained throughout the modelling process. Ideally, all validity conditions of the features, i.e. their semantics, should be checked by the system after each modelling operation. If some validity condition is no longer satisfied, e.g. the top face of a hole is no longer open, this should be notified by the system, and preferably the user should be assisted in overcoming this situation. An approach that supports these ideas is semantic feature modelling (Bidarra and Bronsvoort 2000). Such an approach can guarantee that all design intent once captured in a model is maintained, which brings feature modelling to a level really higher than advanced geometric modelling. In design by freeform features, the creation of individual freeform features by specifying values for their parameters is again the basic idea. Because of the large shape domain of freeform features, a good mechanism to let users define

4 new feature classes, in which the generic shape, the parameterisation and the validity conditions are easy to specify, is indispensable. In Section 4, a technique to realise this for freeform volumetric features is introduced. In addition, ways to indicate how individual freeform features in an object have to be attached to the model, including the order of continuity between adjacent faces of features, have to be provided. Au and Yuen (2000) present a modelling language for freeform objects that covers the latter, but much work remains to be done here. Validity maintenance has hardly been explored for regular-shaped features, let alone for freeform features. As already remarked above, one can easily imagine useful validity conditions for freeform features. However, these become only really useful if they are maintained during the modelling process, e.g. to preserve the design intent once captured in a model or to guarantee manufacturablility of the product. In feature recognition, features are recognised from a geometric model of a product. Historically, this was the first method to identify features in a model, introduced in the context of manufacturing planning. Many methods for feature recognition exist, each with its own advantages and disadvantages (Shah and Mäntylä 1995). The four most important categories of feature recognition methods that can currently be identified are rule-based, graph-based, volume decomposition and geometric reasoning methods. Some successful feature recognition systems use combinations of these methods. Although some methods have been proposed for freeform feature recognition, e.g. by Sonthi et al (1997), Little et al (1998) and Lim et al (2001), these are not yet mature. Considering the long history in, and large variety of currently available methods for, recognition of regular-shaped features, much more research is foreseen here. In Section 5, a new technique for recognising freeform surface features, based on template matching, is introduced. Once a feature model of a product has been created, either by design by features or by feature recognition, other feature models of that product, which correspond to other views, can be derived by feature conversion. For example, a manufacturing planning view can be derived from a design view. Feature conversion is a relatively new technique, and forms the basis for multipleview feature modelling systems (de Kraker et al 1995; Noort and Bronsvoort 2000). In such systems, several views on a product can be simultaneously maintained by the system. Modifications made in one view, are automatically propagated to the other views on the product. This can be used to, among other things, support design for manufacturing: while a model is being built with design features, it is immediately converted into a manufacturing planning view, which can be used to check the manufacturability of the product. Feature conversion and multiple-view feature modelling have not yet been tackled in the context of freeform features. Several topics mentioned above come together here, such as parameterisation of features, validity specification and maintenance, and feature recognition. 3. CLASSIFICATION OF FREEFORM FEATURES To further illustrate the concept of freeform features, some classifications of such features are given. Also some remarks on their parameterisation are made. The set of features useful in a freeform feature modelling system depends on the application domain. The main distinction that can be made is between freeform surface features, which consist of one or more freeform surfaces that do not necessarily bound a volume, and freeform volumetric features, which consist of a volume bounded by a number of freeform surfaces. Until now, freeform features were mostly considered to be surface features, which links up with the principle of freeform surface modelling. Two classification schemes of such freeform surface features are given here. Poldermann and Horváth (1996) provide a general classification of freeform surface features. Four major classes of freeform surface features are introduced, namely primary surface features, modifying surface features, auxiliary surface features and transition surface features (see Figure 2). The primary surface features define the global shape of a product, whereas the modifying surface features represent secondary features that modify the primary surface features. The auxiliary surface feature class includes wellknown mechanical features, such as holes and ribs. The transition surface features, corresponding to blends, are included to be able to ensure continuity at surface boundary connections.

5 Freeform features Primary surface features Closed Single or double curved Amorphous Opened Non-isometric Isometric Modifying surface features Local Protrusion Depression Global Bend Wave Blending Auxiliary surface features Holes Simple Threaded Cuttings Lifted Lowered Ears Single Double Fillets Arc Chamfer Ribs Cross Fix-length Slicing Deformed Non-deformed Transition surface features Simple Blending Rounding Compound Blending Rounding Figure 2. Classification by Poldermann and Horváth Fontana et al (1999) present a freeform feature taxonomy for secondary features, specifically suited for design by features. As shown in Figure 3, the classification contains two types of features, namely δ-features, which are basically deformations of freeform surfaces, and τ- features, which are eliminations of areas in a freeform surface. The two types of features are in fact a subset of the features presented by Poldermann and Horváth, but they are more formally defined. Three types of deformations are distinguished, namely border deformations, internal deformations and n-channel deformations, depending on the placement of the feature on the surface. Every deformation can be directed into the product (intrusion) or out of the product (extrusion). Freeform features Freeform δ-features Border Intrusion (Step-up) Extrusion (Step-down) Internal Intrusion (Cavity) Extrusion (Bump) N-channel Intrusion (N-groove) Extrusion (N-rib) Freeform τ-features Sharp Inlet Hole N-gap Finished Inlet Hole N-gap Figure 3. Classification by Fontana et al Two types of eliminations are listed, namely a sharp cut and a finished cut. The finished cut is actually a combination of a sharp cut and a deformation around the edge of the cut in order to smoothen the cut. Both sharp and finished cuts can be divided into inlets, holes and n-gaps, depending on the resulting connectivity of the surface, respectively simply connected, connected but not simply connected, and disconnected. Poldermann and Horváth (1996) and Fontana et al (1999) in fact present freeform features as ways to modify a given freeform surface. They distinguish adding, deforming (or substituting) and removing regions of such a surface, and their freeform features are related to these operations. As pointed out in Section 2, feature instances are determined by specifying values for the parameters in the corresponding generic feature class. For regular-shaped features, it is relatively easy to think of parameters that uniquely specify the shape of a feature, e.g. height and radius of a cylindrical protrusion. Also, it is easy for a user to understand what the influence of a parameter like height or radius is. It is more difficult to assign intuitive parameters to freeform features. Poldermann and Horváth (1996) stated that at the assignment of parameters to freeform surface features, two requirements

6 should be taken into account. First, the parameters must be related to the parameters preferably used in design, e.g. radius, height and angle. Second, they must be related to the entities of their geometric representation, e.g. the control network, weights and knot vectors of NURBS surfaces. The control points of a NURBS surface generally do not lie on the surface itself, so that when they are manipulated by designers who do not have knowledge of the mathematical background of the representation, these do not perceive their behaviour as intuitive and predictable. Therefore, a high-level representation with parameters that seem logical to the designer is desired. As a designer can only handle a limited number of parameters at a time, the number of parameters assigned to a feature class should be limited (Vergeest et al 2001). In Figure 4, an example of a parameterised freeform feature, a four-sided freeform protrusion, is shown, along with a set of parameters that uniquely define the feature and are easy to comprehend by a user. (a) (b) Figure 4. Freeform protrusion feature with parameters To implement the above, a mapping has to be performed from a set of intuitive parameters to a set of surface patches, which in the case of NURBS are determined by a control network, weights and knot vectors. An unambiguous mapping must be performed, so that for every feature instance whose parameters have been specified, a unique mathematical representation can be generated. As stated earlier, instead of freeform surface features, also freeform volumetric features can be considered. Many important freeform volumetric features, such as protrusions, depressions, ribs and holes, can rather easily be created from their counterpart in the set of freeform surface features, by closing the set of surfaces defining the latter. Obviously, also other types of freeform volumetric features can be considered. The main advantage of using freeform volumetric features, instead of freeform surface features, is that these better link up with regular-shaped features, which are also volumetric features. Since there are many application domains, it is difficult for a modelling system to offer a complete library of freeform features. It is therefore very important that users have a method at their disposal to define their own freeform feature classes, as also stated by Poldermann and Horváth (1996) and Au and Yuen (2000). In the next section, a technique for defining freeform volumetric feature classes and their parameterisation is introduced. 4. DEFINITION OF FREEFORM VOLUMETRIC FEATURE CLASSES At the Computer Graphics and CAD/CAM group of Delft University of Technology, a research project on validity maintenance for freeform feature modelling is being performed. The semantic feature modelling approach described in Section 2 will be extended to freeform feature modelling. In Section 3, a distinction was made between freeform surface features and freeform volumetric features. Here, freeform volumetric features will be considered. Developing a validity maintenance approach is more useful for volumetric than for surface features, because then freeform features can be integrated with the more traditional regular-shaped features, which are also volumetric. In addition, many validity conditions are related to feature volumes, instead of their surfaces only. An important step in extending the semantic feature modelling paradigm involves the definition of generic freeform feature classes. The structure of these classes resembles the structure of generic regular-shaped feature classes (Bidarra and Bronsvoort 2000), as they both contain shape, position and orientation, and validity information (see Figure 5). However,

7 new ways to describe a wider range of shapes are needed in freeform feature modelling. Also, other ways to position and orient freeform features and new validity conditions are required, to match the different geometry and properties of freeform features. Figure 5. Freeform volumetric feature class structure A freeform volumetric feature library manager offers users an interface for interactively defining generic feature classes. The definition process will be illustrated for a freeform rib feature. The process starts with the creation of a prototype feature shape. This can be done using several techniques, such as sweeping and skinning, but also through instantiation and manipulation of a canonical shape, containing one or more freeform faces. Several canonical shapes may be available as templates in a separate library. In Figure 6a, a prototype swept shape is shown; it gives the shape of a freeform rib feature, and is specified by a profile and a path. Next, the prototype shape has to be parameterised, making it suitable to be included in a generic feature class. As already mentioned in Section 3, feature parameters have to be intuitive and their influence has to be predictable. During parameterisation, feature parameters are assigned, directly through geometric constraints or indirectly through algebraic constraints, to the geometric elements with which the prototype has been built. For the swept prototype, a profile and a path curve have been specified, and parameters are assigned to these geometric entities using combinations of geometric and algebraic constraints. Figure 6b shows the prototype from Figure 6a, now fully parameterised. Notice that some parameter values for an instance of the freeform rib feature may be explicitly specified by the user, e.g. the height parameter, whereas other parameters may be implicitly specified by the feature model, e.g. the length parameter dependent on the way the rib is attached to the feature model. As with regular-shaped features, the nature of a freeform feature indicates whether it adds or removes material from a feature model, respectively called additive and subtractive natures. An attach constraint of a feature couples one or more of its faces to a feature face that is already present in the feature model. The planar bottom face of the freeform rib feature defined in this section could be attached to the top face of a block. Attach constraints are currently under investigation, in particular for attaching curved surfaces to each other. Procedural methods, such as sweeping and skinning, could be helpful in this process, e.g. by using some curve on the target surface as the path parameter for a sweep operation, thereby ensuring that the sweep attaches seamlessly to this target surface. Geometric constraints position and orient a feature relative to other features present in the model, by fixing its remaining degrees of freedom. For example, the freeform rib feature could be positioned relative to several reference feature faces in the feature model with distance constraints. Dimension constraints can be used to specify the set of values allowed for a parameter. For example, the width of the freeform rib feature could be limited to values between 5 and 20. Algebraic constraints constitute relations between parameters. They can be simple equalities, e.g. length equals width, but also more general algebraic expressions involving two or more parameters and constants. Curvature constraints can put restrictions to the derivative in any direction at any position on a curved surface. Boundary constraints specify which topological variants of a feature instance are allowed, stating

8 for feature faces whether they are onboundary, meaning that the face should be present on the model boundary, or notonboundary, meaning that the face should not be present on the model boundary. Faces can be constrained to be completely or partly onboundary, or completely or partly notonboundary. For the freeform rib feature class, it might be specified that the top face (see Figure 6b) should be completely onboundary, whereas the front and back faces should be partly onboundary. (a) (b) Figure 6. Definition of a freeform rib feature A higher level of validity specification is achieved with interaction constraints. Feature interactions are modifications of shape aspects of a feature that affect its functional meaning. Examples are disconnection interaction, where a feature volume of an additive feature becomes disconnected from the model, absorption interaction, where a feature volume is completely contained in another feature volume, and splitting interaction, where a feature is split into several parts. Interaction constraints specify whether such interactions are allowed for a feature class. Volume constraints put restrictions on the volume of a feature. For example, minimal and maximal volume values could be specified here. The last constraint type mentioned here is the continuity constraint. Using these constraints, faces within a feature instance or between different feature instances can be set to retain a certain level of continuity, e.g. C 1 continuity. Several other validity constraint types are expected to be useful in relation to freeform volumetric feature classes, and research is therefore ongoing in this area. However, not only new constraint types need to be developed, existing constraint types need to be adjusted too to match the complex shapes in freeform volumetric feature modelling. 5. FREEFORM SURFACE FEATURE RECOGNITION At the Design Engineering group of Delft University of Technology, feature recognition in the domain of freeform shapes is being studied. In particular, feature recognition by template matching is the subject of the research described here. Generally, four possible reasons for performing feature recognition can be identified: a) the product model was created by design by freeform features, but all feature information except the explicit geometry got lost, e.g. during a CAD data exchange process, b) the designer intentionally created the features using low-level geometric modelling techniques, c) the features are a by-product of feature interaction, d) conversion between different modelling views is required. What counts is that features are "shape aspects" that need to be controllable for the designer by means of parameters and constraints that are most appropriate in the situation at hand. The feature parameters need to be selected such that they are perceived as natural and effective by the designer. We can write selecting here, since, theoretically, a particular freeform feature can be parameterised in infinitely many ways (Vergeest et al 2001). Ideally, the choice of parameterisation should be adaptive to the specific design context. In some cases it might be effective to support local control over the model, where the relevant parameters influence only a limited region of the boundary. The scope of the research concerning freeform feature recognition is limited to regions S of surfaces, although the method may be extendable to volumetric features. Formally, S! 3 is a portion of the boundary of a compact subset of! 3. Without losing generality, we assume that S is a 2-manifold in! 3. Likewise, T(p) is a 2- manifold in! 3 and p P is called a parameter value from the parameter domain P, which can be any set. Informally, S and T(p) are called 3D shapes. The mapping T : P 2!3, where 2!3 denotes the power set (the set of subsets) of 2!3,

9 determines a family of shapes {T(p) 2!3 p P}. T is now interpreted as a shape feature type. Furthermore, we define the directed Hausdorff distance from T(p) to S as H ( T ( p), S) = sup ( inf t s ), t T ( p) s S where t - s denotes the Euclidean distance between the points s and t. The mean directed Hausdorff distance M(T(p), S) is 1 M ( T ( p), S) = inf t s da, Area( T ( p)) s S T ( p) where the integration is over the surface of T(p), normalised by the surface area of T(p). The mean directed Hausdorff distance is sometimes preferred over the directed Hausdorff distance, as the latter is sensitive to noise and inaccuracies in the shape data. If S represents the "shape aspect" in a design context where parameters from a space P are required, then we need to match T(p) to S, which is an optimisation problem of the form Popt = Arg min M ( T ( p), S). p P The parameter space P will contain the space of rigid body transformations (6 degrees of freedom) and one or more intrinsic shape parameters, e.g. to deform T(p). In case of a rib, we need at least the parameters that control the height (h) and the width (w) of the rib. In Figure 7, some variations are visualised. efficiency reasons, in all search strategies the amount of data entering the computation was progressively increased during subsequent stages of the process. The fitting strategies differ in the fraction of the points that actually enter the fit, in the number of stages in which the total procedure is divided, and in the grouping of different parameters that are fitted concurrently. One semiautomatic and three full-automatic fitting strategies were designed. Figure 8 shows a snapshot of the fitting routine at work. Figure 8. Only a portion of the scanned data from a physical object was used to represent S. The points on the template T(p) are also shown Experiments revealed that the semi-automatic approach was the most successful. Some user intervention was sometimes required to move T(p) into the proximity of the rib; from then on the software could fine tune position, orientation and intrinsic shape. In Figure 9, the results of the experiments concerning the four fitting strategies, of which A is the semi-automatic approach and B, C and D are the full-automatic approaches, are shown in a graph. Figure 7. A rib template is represented by points on T(p). Variations of T(p) are shown for different values of the intrinsic shape parameters h and w Shape matching software was developed to search for P opt. The search problem was approached by applying four different fitting strategies (Spanjaard and Vergeest 2001). For Figure 9. The first 100 seconds of the mean directed Hausdorff dissimilarity M(T(p),S) as function of elapsed time More experimentation is needed to better understand the matching procedures, and to make them more efficient. Also the embedding of the feature matching into the design workflow, and the integration of partial surface features with freeform volumetric features needs further work.

10 6. CONCLUSIONS In this paper, first an overview of the current status in freeform feature modelling has been given. Next, two techniques have been described that deal with open topics within the field. First, a definition process for freeform volumetric feature classes has been introduced. A volumetric approach was chosen because many validity conditions are related to feature volumes, instead of their surfaces only. Then, it was shown that introducing freeform geometry to feature modelling requires the development of new ways to parameterise freeform shapes and new constraint types. When such constraint types are incorporated in a freeform volumetric feature modelling approach, designers can dispose of a powerful modelling mechanism. Second, some advancements in the parameterisation, computation and matching in the domain of freeform shapes were presented. These results have been tested for surfaces, and are not yet valid for volumetric features. Freeform feature modelling is a relatively new research area. Results from the projects described in this paper show, however, that there are good prospects in this field. Freeform features obviously extend the modelling domain of the systems that incorporate them. However, much research is still to be done before freeform feature modelling is mature. ACKNOWLEDGEMENTS Eelco van den Berg s work is supported by the Netherlands Organization for Scientific Research (NWO). REFERENCES Au C.K. and Yuen M.M.F., (2000), A semantic feature language for sculptured object modelling, Computer-Aided Design, Vol. 32, No. 1, pp Bidarra R. and Bronsvoort W.F., (2000), Semantic feature modelling, Computer-Aided Design, Vol. 32, No. 3, pp Bronsvoort W.F. and Jansen F.W., (1993), Feature modelling and conversion key concepts to concurrent engineering, Computers in Industry, Vol. 21, No. 1, pp Fontana M., Giannini F. and Meirana M., (1999), A free form feature taxonomy, Proceedings Eurographics '99, Brunet P. and Scopigno R. (eds), Computer Graphics Forum, Vol. 18, No. 3, pp de Kraker K.J., Dohmen M. and Bronsvoort W.F., (1995), Multiple-way feature conversion to support concurrent engineering, Proceedings Solid Modeling '95 - Third Symposium on Solid Modeling and Applications, May, Salt Lake City, USA, Hoffmann C.M. and Rossignac J. (eds), ACM Press, New York, pp Hoffmann C.M. and Joan-Arinyo R., (1998), On user-defined features, Computer-Aided Design, Vol. 30, No. 5, pp Lim T., Corney J. and Clark D., (2001), A laminae approach to constructing geometric feature volumes, Proceedings Solid Modeling '01, Sixth ACM Symposium on Solid Modeling and Applications, 6-8 June, Ann Arbor, USA, Anderson D.C., and Lee K. (eds), ACM Press, New York, pp Little G., Clark D.E.R., Corney J.R. and Tuttle J.R., (1998), Delta-volume decomposition for multisided components, Computer-Aided Design, Vol. 30, No. 9, pp Noort A. and Bronsvoort W.F., (2000), Enhanced multiple-view feature modelling, CAD Tools and Algorithms for Product Design, Brunet P., Hoffmann C. and Roller D. (eds), Springer, Berlin, pp Piegl L.A. and Tiller W., (1997), The NURBS Book (Second Edition), Springer Verlag, Berlin. Poldermann B. and Horváth I., (1996), Surface based design based on parametrized surface features, Proceedings International Symposium on Tools and Methods for Concurrent Engineering, May, Budapest, Hungary, Horváth I. and Varadi K. (eds), Institute of Machine Design, Budapest, pp Shah J.J. and Mäntylä M., (1995), Parametric and Feature-based CAD/CAM; Concepts, Techniques and Applications, John Wiley & Sons, New York. Sonthi R., Kunjur G. and Gadh R., (1997), Shape feature determination using the curvature region representation, CD-ROM Proceedings Solid Modeling '97 - Fourth Symposium on Solid Modeling and Applications, May, Atlanta, USA, Hoffmann C.M. and Bronsvoort W.F. (eds), ACM Press, New York, pp Spanjaard S. and Vergeest J.S.M., (2001), "Comparing different fitting strategies for matching two 3D point sets using a multivariable minimizer", Proceedings of the 2001 Computers and Information in Engineering Conference, 9-12 September, Pittsburgh, USA, ASME, New York, DETC'01/CIE Vergeest J.S.M., Horváth I. and Spanjaard S., (2001), "Parameterization of freeform features", Proceedings Shape Modelling International 2001 International Conference on Shape Modelling and Applications, 7-11 May, Genoa, Italy, Pasko A. and Spagnuolo M. (eds), IEEE, Piscataway, pp