A Hybrid Shape Representation: Surface Modeling - Research Proposal Example

 A Hybrid Shape Representation: Surface Modeling Surface Modeling Introduction Computer aided design and engineering has grown to become pivotal to everyday life. Today, almost everything humans use could have design with the aid of a computer application. What began as a rather confined area of design has grown to become expansive and today, multiple approaches and software are used in design. One of the techniques which have gained widespread use across the globe is surface modeling. Today, almost every designer acknowledges the importance of surface modeling in computer aided design. As a matter of fact, Computer Aided Design has become an integral and indispensible part of design, analysis, and production of gadgets across the globe. The number of CAD systems has magnanimously risen over the recent past. Four categories described the systems, which are Wireframe CAD, Surface CAD, 2D CAD, and Solid CAD. Nonetheless, this paper narrows down its focus to solely Surface modeling although in a number of instances appropriate comparisons, more especially to solid modeling will be considered. Before delving deep in the subject, it if vital to highlight what surface modeling is all about. Surface modeling defined According to Allègre et al. (2010), surface modeling generates a model which has minimally ambiguous representation as compared to wireframe although non-comparable to solid modeling which often provides and increasingly realistic perspective. Surfaces and curves are often at the center of their construction. This is a definitional approach accepted by (Lin, Ball, & Zheng, 2008) who nonetheless simplifies it by defining surface modeling as a component’s skin, inclusive of the surfaces and edges. Put in another way, surface models are three-dimensional (3D) models which lack thickness (Tobias, 2009). An important thing which should be emphasized about surface modeling is its representation of not just the surfaces and edges but the shape design as well. Engineers define as surface as mathematical presentation of the skin of a component. Generally, B-Splines and Beizer mathematical methods are common tools in controlling curves when undertaking surface modeling. Basically, surface models have neither thickness nor volume and there construction starts with construction of wireframe entities which are then corrected appropriately using a relevant surfacing technique. A sample surface model illustrated below: Figure 1: Sample drawing of a surface model (Allègre et al., 2010) It is important not to confuse surface models with thick models which have some mass properties in them. As earlier mentioned, surface models have no thickness at all, as compared to the thick and solid models which have some thickness defined by the users? Normally, the same tools are used to create surface and solid models. Any solid model which can be created by modeling tool can also have its surface model generated by the same tool. What differs between solid and surface models is the former has mass properties while the latter does not. In some instances, complex shapes cannot be created via solid modeling (Sederberg & Parry, 2008). Nonetheless, such can be created using surface modeling after which they are converted into solid models. It should be noted that persons who are familiar with solid modeling find it easier to learn surface modeling. Various types of surface models In computer aided design systems, there are various surface types including the curved surfaces, the flat plane surfaces and the fillet surfaces. Each has distinct features which make it different from the others. There are a number of curved surfaces including Surfaces which are attached to data point arrays, curved Mesh or Sculptured Surfaces, and of course, others related to curves such as the ruled surface, the tabulated cylinder, swept surfaces, as well as the surface of revolutions. Surfaces attached to data point arrays as also referred to as control points (Lin, Ball, & Zheng, 2008). They are produced by passing via or interpolating a number of points. This is unlike the curve mesh. Sculptured surfaces which are produced through generation of grid curves which form surface patch works. Flat Plane Surfaces on the other hand are defined in a number of ways such as via triple points or through a point and a line or via lines which lie in parallel while fillet surfaces refer to a curve which is interpolated between a number of surfaces, for instance, the chamfer surface. Advantages and disadvantages of surface modeling Like other modeling techniques, surface modeling has lots of benefits to offer and so does it have some shortcomings. Firstly, it is important to mention that unlike many modeling techniques, it is less ambiguous, allows creation and identification of complex surfaces, in addition to being able to remove hidden line and also add some level of realism (Sederberg & Parry, 2008). Nonetheless, surface model are not easy to construct, makes it difficult to calculate a model’s mass property let alone mention that creating them requires more time. The storage space is also larger as compared to the others. These will however be addressed later when looking at the improvements which can be made on this modeling technique to make it even better. Solid versus surface modeling Perhaps one may ask, why model the surface when you can model the real thing? While this question may find meaning in the eyes of a layman, a designer, an engineer or an architect may take and entirely different perspective. While solid modeling presents a realistic representation of the object, it lacks some elements vital to the aforementioned. It must be emphasized that Computer aided design (CAD) is not comparable to a car which can be used quite well irrespective of the knowledge on its operation and mode of working (Igarashi & Matsuoka, 2010). Computer aided design offer an opportunity to understand what is happening beyond the naked eye. Understanding surface modeling ultimately affects your understanding of the model deeper. As Allègre et al. (2010).puts it, surface and solids are just but underlying mathematics which defines the created forms geometry. As already mentioned, 3D geometry can be defined in three forms including solids, surfaces and wireframes. Wireframes have a little or rather negligible role to play in CAD but are rather the basis of digital content creation and gaming. The simplest and most appropriate means of understanding surface and solids modeling variations is to visualize a water balloon whereby the water in the balloon defines solids modeling while latex skin used to make the balloon defines surface model. Explaining further, solid models are objects which have a geometric mass and are normally created by addition or subtraction of subsequent features to a base solid. This modeling approach employs features such as extrusion, extrusion cuts, revolving, radii and chamfers. On the contrary, surface modeling defines exterior of an object using an infinitesimally thin skin. This is created using lofts, sweeps, and curves (Igarashi & Matsuoka, 2010). In essence, their creation involves surfaces which are sculptured and as such have lots of curvatures. These surfaces are either pole or guide curve defined. What makes a distinction between a solid and a surface is the enclosure. Surface modeling is mainly used in generation of complex or rather technical surfaces. Development of this technique in the 70’s was mainly in response to the needs of aerospace and the automotive industry. The approach is often considered more difficult as compared to solid modeling although they are more robust. More often, if not always, solids modeling programs use surface modeling whereby the solids are generated through creation of a surface, which is then filled to come up with a solid (Sederberg & Parry, 2008). Quite often, experts have argued that surface modeling applied in solids yields unstable geometries which are easily susceptible to collapse and failure. In essence, the fact that a change in one dimension is likely to cause a subsequent change in another dimension makes it a vulnerable model. Understanding the basics of surface modeling As has been mentioned in the earlier sections, surface modeling is useful in generation of sophisticated shapes in 3D CAD systems. The models are amazing when it comes to creation of shapes where product form is accorded high priority. The technique is useful in design in consumer product designs in instances where ergonomics and aesthetics of the product are top priority and hence attained using sophisticated shapes and curves (Igarashi & Matsuoka, 2010). The models can then assist in making of molds, rapid prototyping, creation of marketing images and any other appropriate need. This section dissects surface modeling and hence shedding more light on their creation. The main types of surface modeling are addressed including: Curved surfaces Polygon surfaces Volumes Polygon Surfaces These refer to a combination of surface polygons which enclose and object’s interior. An illustration is provided in the diagrams below: Figure 2: Polygon surface model (Igarashi & Matsuoka, 2010). Polygon surfaces are specified using vertex coordinates as well as using associated attribute parameters. This is illustrated below: Figure 3: vector representation surface modeling (Igarashi & Matsuoka, 2010). Polygon meshes have a set of characteristics which make them distinct from other forms of surface models. These are highlighted below. The meshes have a set of connected planar surfaces which are bound by polygons They are appropriate for boxes, cabinets, and building exteriors, among others. They are inappropriate where curved surfaces are required. They are susceptible to arbitrarily small errors at the expense of space and period of execution. Their enlarged images reveal geometric liaising. Other than the things mentioned, polygons meshes are appropriate in representation of flat-faced objects mentioned, non-satisfactory when it comes to representation of curved surface objects (Lin, Ball, & Zheng, 2008). They experiment space inefficiencies, and also have simpler algorithms. It is however held that polylines and polygons all require extensive amount of data in order to achieve accuracy. Their interaction manipulation is as well very tedious. Curved surfaces Curve representations can either take the form of explicit, implicit, or parametric forms/functions. For high order curves, there is more ease in manipulation in addition to being more compact and hence requiring lesser storage space. Typical curves are generally defined in terms of Cartesian planes/coordinates as shown; Explicit functions Explicit functions are defined as shown below, In this case, it is not possible to obtain multiple single x values and hence curves are broken down into circle and ellipse segments. They are additionally non-invariant with rotations which might require additional breaking of segments (Sederberg & Parry, 2008). These however pose a problem when dealing with curves which have vertical tangents and infinite slopes are not easy to represent. Implicit functions Unlike the explicit functions, these are defined by the equation below; Such equations can be solved when one so desires for instance x² + y² = 1, half circle. However, these pose a problem when it comes to joining of the curves together. It is further very difficult to establish if the tangent’s directions are in agreement at the joint points. Parametric curves The curved surfaces are largely motivated by the need to achieve precise and more concise representation of objects. Often, a good appropriate surface representation should incorporate a number of traits including accuracy, concision, intuitive specification, arbitrary topology, guaranteed continuity, efficient intersections and display, as well as local support (Sederberg & Parry, 2008). Narrowing down to parametric surfaces, these are bound using parametric functions for example, A good example to illustrate these functions is the ellipsoid which is represented as follows; While for parametric surfaces it is easy to enumerate surface points, there is often a need for piecewise-parametric surface in order to describe sophisticated shapes and surfaces. See diagram below: Figure 4: Parametric representations Figure 5: Piecewise parametric representations (Requicha & Voelcker, 2011). It is however important to mention that these parametric polynomials overcome the challenge associated with explicit and implicit surface forms and equations. They have no geometric slopes but rather have non-infinite parametric tangent vectors (Sederberg & Parry, 2008). The polynomial curve equations x (t), y (t), z (t) are usually cubic curves. Outlined below are additional factors which define parametric curves. Patch boundaries are defined by parametric polynomial curves They have more complex algorithms as compared to the polygon meshes. Parametric patches are also fewer as compared to polynomial patches when approximating curved surfaces to a specified level of accuracy Figure 6: Parametric patches Many people have often questioned why cubic curves are used. Well, they help in handling a lot of challenges which face modeling. Firstly, it is important to mention that low-degree polynomials offer little flexibility when it comes to control of the curve’s shape (Sederberg & Parry, 2008). On the other hand, high-degree polynomials may introduce undesirable wiggles and create a requirement for additional computations. The low end degrees allow end-point specification as well as their derivatives, in addition to not being planar in 3D form. Kind of continuity offered includes: G0: two joined curve segments G1: tangent’s directions equal at the joint C1: tangent directions as well magnitudes equal at the joint Cn: directions and magnitudes of nth derivative equal at the joint The major are discussed hereafter; Hermit: these are defined using two end-points alongside two tangent vectors. Bezier: these are defined by two end-points and two additional points which control tangent vectors end-point (Lin, Ball, & Zheng, 2009). The Bezier bi-cubic formulations are as shown below: Figure 7: Sample Bezier curve (Lin, Ball, & Zheng, 2009). Splines: there are various kinds with each defined by four points (Barghiel, 2009). They are uniform B-splines, non-uniform B-splines, or ß-splines. It bi-cubic formulation is as shown below: Discussion of solid modeling As has been illustrated in the previous sections, surface modeling is a sophisticated modeling approach which calls for vast technical knowledge if one is to successfully use it in representations of items. Nonetheless, it must be appreciated that this techniques brings on board lots of benefits to the modeling world. Firstly is its ability to eliminate unnecessary ambiguity and lack of uniqueness. This is very common with wireframe models which hide unseen lines. Additionally, surface modeling enhances a model’s visualization as well as presentation and hence presents an increasingly realistic view. It cannot also be forgotten that computer aided manufacturing requires surface geometry, something which is offered by surface modeling. This technique is therefore important in manufacturing as it offers a manufacturing appropriate geometry. For instance, engineer’s intent on molding or die designs will find this an appropriate avenue to use. Additionally, the technique allows design and analysis of complex shapes not achievable by other forms of modeling. Ship hulls, airplane fuselages as well as car bodies are some of the beneficiaries of this modeling approach. Other than the aforementioned, this is a technique which allows surface properties including roughness, color as well as reflectivity to be assigned, evaluated and sufficiently demonstrated. It is also usable in approximation of approximate intersections. Nonetheless, it cannot be claimed to have yielded perfection in modeling advances. Like other modeling techniques, surface modeling is not without troubles. There are a number of challenges which have been recorded over the past in relation to surface modeling. Surface modeling offers no information at all with regard the internal structure of an object. This in essence implies that for sufficient communication between technocrats, this model cannot be used alone. It must be accompanied by other modeling approaches. Additionally, their accuracy is dependent on having a fine mesh and in instances where the meshing is coarse, wrong results may be obtained (Lin, Ball, & Zheng, 2008). Nonetheless, the most challenging thing about surface models is that they entail complex computations based on the number of surfaces being generated. NC programmers also find surface modeling quite challenging to use. A lot of proficiency is necessary in order to successfully implement and basically, it takes more than simple surface mathematics to successfully deploy the technique. Although it can be learnt, machinists often find it quite difficult and challenging. Perhaps, it is time researchers in this area explore how these technique can be made more user friendly and detached from the need for extensive technical knowledge. The complexity associated with finished surface models also creates a challenge to user-friendliness of this model. Also challenging is the inconsistency associated with individual orientation of surfaces. Whilst some surfaces may need to face up, the neighboring ones may need to face downwards. This causes a problem with machine algorithms considering that orientation has a role in determination of the surfaces and surface parts which are to be machined. Transfer of the resulting files across the various surface modeling tools also poses a challenge. While interactive system demonstrates the ability of surface modeling, there is an expansive space for improvement. A key limitation highlighted is the application of discrete exponential maps in parameterization of the edit regions considering they regularly break down when traversing high curvature regions. Nonetheless, by developing robust algorithms wobbling below the underlying mesh can be addressed, although the distortion in the DEM maintains consistency and hence produce more rigid structures (Lin, Ball, & Zheng, 2008). In essence, increased parameterization algorithms have potential of increasing the system range enormously. Another limitation worth improving is the response time of systems which often varies depending on the edit being manipulated, given that update depends on the depth of the modified layer. Nonetheless, computation of edits at lower resolutions in case of interactions can solve such challenges although fidelity loss may not be desirable. Conclusion The basic goal of this research was to evaluate, understand and on this basis suggest improvements to the power of surface modeling tools which can be used by designers. Amongst the proposals arrived at include creating mechanisms which help designers to go back in time and modify modeling from the past. The underlying hierarchy of procedures allows effective exploration of design variations without the need to discard the work already in existence. Additionally, allowing for linked copy-and paste options allows and make easier performance of repetitive modeling task. Additionally, a platform should and need to be created for combination of linear variables with parametric variables in order to come up with appropriate composite functions relevant to design. In comparison to regular geometric modeling techniques, this approach can be used in representation of complicated tools for surface modeling with a single patch and hence effectively help in surface manipulation. In essence, modeling of the items is inevitable. Therefore, inspection, monitoring, modeling, restoration, as well as preservation of automatic and more precise techniques are urgent to the modern society. References Allègre R. et al. (2010). A hybrid shape representation for free-form modeling. Shape Modeling International, 34, pp. 123 – 134. Barghiel C. (2009). Spline surfaces. Mathematical Methods for Curves and Surfaces, pp. 31–40. Igarashi T. & Matsuoka S. (2010). Sketching interface for 3d freeform design. SIGGRAPH, 99, pp. 409–416. Lin, J., Ball, A.A. & Zheng, J.J. (2008). Surface modeling and mesh generation for simulating superplastic forming. Journal of Materials Processing Technology, 80–81, pp. 613–619. Lin, J., Ball, A.A. & Zheng, J.J. (2009). Approximating circular arcs by Bezier curves and its application to modeling tooling for FE forming simulations. International Journal of Machine Tools & Manufacture, 41, pp. 703–717. Requicha A. & Voelcker H. B. (2011). Solid modeling: Current status and research directions. IEEE Comp. Graph & Appl. 3, pp. 25–37. Sederberg T. W. & Parry S. R. (2008). Free-form deformation of solid geometric models. SIGGRAPH, 20, pp. 151–160. Tobias, F. (2009). 3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data. Computers & Geosciences, 33, pp. 932–943 Read More