Boundary Representation Modelling Techniques

Representations for solid modelling. 3. Cell decomposition. 3. General sweeping. •. 3. Set theoretic. 5. Boundary representation. 5.
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No isolated or dangling boundaries see Figure2 should be permitted. Finiteness and finite describability. The former property means that the size of the solid is not infinite while the latter ensures that a limited amount of information can describe the solid. The latter property is needed in order to be able to store solid models into computers whose storage space is always limited.

It should be noted that the former property does not include the latter and vice versa. For example, a cylinder that may have a finite radius and length may be described by an infinite number of planar faces. Closure under rigid motion and regularized Boolean operations. This property ensures that manipulation of solids by moving them in space or changing them via Boolean operations must produce other valid solids.

The boundary of a solid must contain the solid and hence must determine distinctively the interior of the solid. Various representation schemes have been designed and developed, with the above properties in mind, to create solid models of real objects. Nine schemes can be identified. Some of them are more popular than the others. These are halfspaces, boundary representation B-rep , constructive solid geometry CSG , sweeping, analytic solid modeling, cell decomposition, spatial enumeration, octree encoding, and primitive instancing. The three most popular schemes are B-rep, CSG, and sweeping.

Representations of solids are built and invoked via algorithms. An algorithm is a procedure that takes certain input and produces a desired output. Algorithms can be classified into three types according to their input and output. Some algorithms take data and produce representations ; that is, a: Representation schemes mentioned above fall into this type. The other type of algorithms compute property values by taking a representation and producing data; that is, a: All application algorithms belong to this type.

For example, a mass property algorithm takes a solid model representation and produces volume, mass, and inertial properties. Algorithms of the third type take representations and produce representations; that is, a: For example, an algorithm that converts CSG to B-rep or one that simulates models processes such as motion or machining on objects belongs to this type.

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An algorithm might take a piece of stock and end up with a machined part. Check out your CAD software. Are there any algorithms compute property values by taking a representation and producing data? Are there any algorithms take representations and produce representations? Boundary representation is one of the two most popular and widely used schemes to create solid models of physical objects.

A B-rep model or boundary model is based on the topological notion that a physical object is bounded by a set of faces. These faces are regions or subsets of closed and orientable surfaces. A closed surface is one that is continuous without breaks. An orientable surface is one in which it is possible to distinguish two sides by using the direction of the surface normal to point to the inside or outside of the solid model under construction.

Each face is bounded by edges and each edge is bounded by vertices. Thus, topologically, a boundary model of an object is comprised of faces, edges, and vertices of the object linked together in such a way as to ensure the topological consistency of the model. The database of a boundary model contains both its topology and geometry. Topology is created by performing Euler operations and geometry is created by performing Euclidean calculations.

Euler operations are used to create, manipulate, and edit the faces, edges, and vertices of a boundary model as the set Boolean operations create, manipulate, and edit primitives of CSG models. Euler operators , as Boolean operators, ensure the integrity closeness, no dangling faces or edges, etc. They offer a mechanism to check the validity of these models. Geometry includes coordinates of vertices, rigid motion and transformation translation, rotation, etc.

It should be noted that topology and geometry are interrelated and cannot be separated entirely. Both must be compatible otherwise nonsense objects may result. Figure3 s hows a square which, after dividing its top edges by introducing a new vertex, is still valid topologically but produces a nonsense object depending on the geometry of the new vertex. In your CAD software, can you build a block strictly using boundary representation? That is, define 8 vertices, connect them to form 12 edges, then define 6 closed, orientable surfaces to form the block.

Render the block to show that your block is defined correctly. If a solid modeling system is to be designed, the domain of its representation scheme objects that can be modeled must be defined, the basic elements primitives needed to cover such modeling domain must be identified, the proper operators that enable the system users to build complex objects by combining the primitives must be developed, and finally a suitable data structure must be designed to store all relevant data and information of the solid model.

Boundary representation

Objects that are often encountered in engineering applications can be classified as either polyhedral or curved objects. A polyhedral object plane-faced polyhedron consists of planar faces or sides connected at straight linear edges which, in turn, are connected at vertices. A cube or a tetrahedron is an obvious example. A curved object curved polyhedron is similar to a polyhedral object but with curved faces and edges instead.

The reader might have jumped intuitively to the conclusion that the primitives of a B-rep scheme are faces, edges, and vertices. This is true if we can answer the following two questions. First, what is a face, edge, or a vertex? Second, knowing the answer to the first question, how can we know that when we combine these primitives we would create valid objects? Polyhedral objects can be classified into four classes. The first class is the simple polyhedra. The second class Figure4 b is similar to the first with the exception that a face may be bounded by more than one loop of edges.

The third class Figure4 c includes objects with holes that do not go through the entire object. The fourth and the last class includes objects that have holes that go through the entire objects. Topologically, these through holes are called handles. With the above physical insight, let us define the primitives of a B-rep scheme. A vertex is a unique point an ordered triplet in space. An edge is a finite, non-self-intersecting, directed space curve bounded by two vertices that are not necessarily distinct.

A face is defined as a finite connected, non-self-intersecting, region of a closed oriented surface bounded by one or more loops.

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A loop is an ordered alternating sequence of vertices and edges. A loop defines a non-self-intersecting, piecewise, closed space curve which, in turn, may be a boundary of a face. In Figure4 a , each face has one loop while the top and the right side faces of the object shown in Figure4 b have two loops each one inner and one outer. A handle or through hole is defined as a passageway that pierces the object completely. The topological name for the number of handles in an object is genus.

The last item to be defined is a body sometimes called a shell. It is a set of faces that bound a single connected closed volume. Thus a body is an entity that has faces, edges, and vertices. A minimum body is a point. Topologically this body has one face, one vertex, and no edges. The object on the right of Figure4 c has two bodies the exterior and interior cubes and any other object in Figure4 has only one body.

Faces of boundary models possess certain essential properties and characteristics that ensure the regularity of the model; that is, the model has an interior and a boundary. Faces are two-dimensional homogeneous regions so they have areas and no dangling edges. In addition, a face is a subset of some underlying closed oriented surface. At each point on the face, there is a surface normal N that has a sign associated with it to indicate whether it points into or away from the solid interior. In traversing loops, the edges of the face outer loop is traversed, say, in a counterclockwise direction and the edges of the inner loops are traversed in the opposite direction, say the clockwise direction.

Euler in proved that polyhedra are topologically valid if they satisfy the following equation: Open objects satisfy the following Euler's law: The boundary model of a sphere has one face and one vertex. We now turn from polyhedral objects to curved objects such as cylinders and spheres.

As shown in Figure 6, the boundary model of a cylinder has three faces top, bottom, and cylindrical face itself , two vertices, and three edges connecting the two vertices. They are called silhouette edges. Show that all 9 objects of Figure 4 satisfy the Euler law in Equation 1. Solid modeling software also has the following five characteristics to enhance the efficiency and productivity of the solid modeling process: In feature-based modeling, a solid model is created and modified in a way that represents how geometries are created using common manufacturing processes.

A base feature is a solid model that is roughly the size and shape of the part that is to be modeled. It can be thought of as the initial work block. All subsequent features reference the base feature either directly or indirectly. Additional features shape or refine the base feature. Features in a part have a direct analogy to geometries that can be manufactured or machined. This gives the engineer the ability to easily create and modify common manufactured features.

As a result, planning the manufacture of a part is facilitated by the correspondence between the features and the processes required to make them.

Boundary representation

Constraint-based modeling permits the engineer or designer to incorporate? Often this is referred to as design intent. The initial sketch of a two-dimensional profile in constraint-based solid modeling does not need to be created with a great deal of accuracy. It just needs to represent the basic geometry of the cross section.

The exact size and shape of the profile is defined through assigning enough parameters to fully? For example, if a hole is constrained to be at a certain distance from an edge, it will automatically remain at that distance from the edge, even if the edge is moved. If the edge were moved, the hole location would need to be respecified so that the hole remains the same distant from the edge.

The advantage of constraint-based modeling is that the design intent of the engineer remains intact as the part is modified. There are two primary types of constraints. Dimensional constraints are used to specify distances between items in a solid model. Geometric constraints define positional relationships between entities in the model in terms of the geometry. Examples of geometric constraints include tangency, parallelism, symmetry, concentricity, and so on. Typically, a combination of dimensional constraints and geometric constraints are used to fully?

Create a simple solid model, a base feature with a through hole for example, using your CAD software. Explore and describe the feature-based and constraint-based modeling features in your CAD software. Parametric modeling means that parameters of the model may be modified to change the geometry of the model. A dimension is a simple example of a parameter. When a dimension is changed, the geometry of the part is updated. Thus, the parameter drives the geometry. An additional feature of parametric modeling is that parameters can reference other parameters through relations or equations.

The power of this approach is that when one dimension is modified, all linked dimensions are updated according to specified mathematical relations, instead of having to update all related dimensions individually. The last aspect of solid modeling is that the order in which parts are created is critical.

This is known as history-based modeling. For example, a hole cannot be created before a solid volume of material in which the hole occurs has been modeled. If the solid volume is deleted, then the hole is deleted with it. This is known as a parent-child relation. The child hole cannot exist without the parent solid volume existing first. Parent-child relations are critical to maintaining design intent in a part. Most solid modeling software recognizes that if you delete a feature with a hole in it, you do not want the hole to remain floating around without being attached to the feature.

Consequently, careful thought and planning of the base feature and initial additional features can have a significant effect on the ease of adding subsequent features and making modifications. The associative character of solid modeling software causes modifications in one object to? For instance, suppose that you change the diameter of a hole on the engineering drawing that was created based on your original solid model. The diameter of the hole will be automatically changed in the solid model of the part, too.

In addition, the diameter of the hole will be updated on any assembly that includes that part. Similarly, changing the dimension in the part model will automatically result in updated values of that dimension in the drawing or assembly incorporating the part. This aspect of solid model software makes the modification of parts much easier and less prone to error. Using the same solid model created in Assignment 1, explore and describe the parametric, history-based feature-based parent-child relation and associative modeling features in your CAD software.

As a result of being feature based, constraint based, parametric, history based, and associative, modern solid modeling software captures? This comes about because the solid modeling software incorporates engineering knowledge into the solid model with features, constraints, and relationships that preserve the intended geometric relationships in the model. The properties that a solid model or an abstract solid should capture mathematically can be stated as follows: This implies that the shape of a solid model is invariant and does not depend on the model location or orientation in space.

Solid boundaries must be in contact with the interior. No isolated or dangling boundaries see Figure2 should be permitted. The former property means that the size of the solid is not infinite while the latter ensures that a limited amount of information can describe the solid. The latter property is needed in order to be able to store solid models into computers whose storage space is always limited.

It should be noted that the former property does not include the latter and vice versa. For example, a cylinder that may have a finite radius and length may be described by an infinite number of planar faces. This property ensures that manipulation of solids by moving them in space or changing them via Boolean operations must produce other valid solids.

The boundary of a solid must contain the solid and hence must determine distinctively the interior of the solid. Various representation schemes have been designed and developed, with the above properties in mind, to create solid models of real objects. Nine schemes can be identified. Some of them are more popular than the others. These are halfspaces, boundary representation B-rep , constructive solid geometry CSG , sweeping, analytic solid modeling, cell decomposition, spatial enumeration, octree encoding, and primitive instancing.

The three most popular schemes are B-rep, CSG, and sweeping. Representations of solids are built and invoked via algorithms. An algorithm is a procedure that takes certain input and produces a desired output. Algorithms can be classified into three types according to their input and output.


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Some algorithms take data and produce representations ; that is, a: Representation schemes mentioned above fall into this type. The other type of algorithms compute property values by taking a representation and producing data; that is, a: All application algorithms belong to this type.

For example, a mass property algorithm takes a solid model representation and produces volume, mass, and inertial properties. Algorithms of the third type take representations and produce representations; that is, a: For example, an algorithm that converts CSG to B-rep or one that simulates models processes such as motion or machining on objects belongs to this type. An algorithm might take a piece of stock and end up with a machined part.

Check out your CAD software. Are there any algorithms compute property values by taking a representation and producing data? Are there any algorithms take representations and produce representations? Boundary representation is one of the two most popular and widely used schemes to create solid models of physical objects. A B-rep model or boundary model is based on the topological notion that a physical object is bounded by a set of faces.

These faces are regions or subsets of closed and orientable surfaces. A closed surface is one that is continuous without breaks. An orientable surface is one in which it is possible to distinguish two sides by using the direction of the surface normal to point to the inside or outside of the solid model under construction. Each face is bounded by edges and each edge is bounded by vertices.

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Thus, topologically, a boundary model of an object is comprised of faces, edges, and vertices of the object linked together in such a way as to ensure the topological consistency of the model. The database of a boundary model contains both its topology and geometry. Topology is created by performing Euler operations and geometry is created by performing Euclidean calculations. Euler operations are used to create, manipulate, and edit the faces, edges, and vertices of a boundary model as the set Boolean operations create, manipulate, and edit primitives of CSG models.

Euler operators , as Boolean operators, ensure the integrity closeness, no dangling faces or edges, etc. They offer a mechanism to check the validity of these models. Geometry includes coordinates of vertices, rigid motion and transformation translation, rotation, etc. It should be noted that topology and geometry are interrelated and cannot be separated entirely.