Curve, Surface And Solid

Curve, Surface and Solid

Curve

Curve Entities • All existing CAD systems provide users with curve entities, which can be divided into analytic and synthetic entities. Analytic entities are points, lines, arcs and circles, fillets and chambers, and conies (ellipses, parabolas, and hyperbolas). Synthetic entities include various types of spline (cubic spline and B-spline) and Bezier curves.

Tables 6.1 to 6.5 show the most common methods utilized by CAD systems to create curve entities

Surface

• Shape design and the representation of complex objects such as car, ship, and airplane bodies as well as castings cannot be achieved utilizing the curves . In such cases, surfaces must be utilized to describe objects precisely and accurately. We create surfaces, and then we use them to cut and trim solid features and primitives to obtain the models of the complex objects. Surface creation usually begins with data points or curves. • Surface creation on CAD/CAM systems usually requires curves as a start. A surface might require two boundary curves, as in the case of a ruled surface that we cover in this chapter. All curves can be used to generate surfaces. In order to visualize surfaces on a computer screen, a mesh, say m x n in size, is usually displayed. The mesh size is controllable by the user.

Surface Entities • During surface creation on a CAD/CAM system, you should follow the modeling guide­lines and strategies. Moreover, you should be careful when selecting curves to create surfaces. Selecting the mismatching ends of curves results in twisted surfaces as shown in Figure 7.1. The figure shows how the wrong ruled surface is created if its defining curves are selected near the wrong ends. The +'s in the figure indicate the selection locations. In such a case, the user deletes the surface and re­creates it by selecting the matching ends. As a general rule, a CAD system uses the midpoint of a curve to interpret the user's click on a curve. If he click is on the right half of the curve, its right end point is selected, and vice versa.

Figure 7.1 Construction of improper and proper surfaces.

Visualization of a surface is aided by the addition of artificial fairing lines (called mesh), which crisscross the surface and so break it up into a network of interconnected patches. The default setting of a CAD system does not display a surface mesh — the surface is displayed with its four boundary curves only. In such a case, the mesh size is 2 x 2, (All surfaces that we create define rectangular patches.) We can change the default mesh size. CAD systems provide users with a menu that allows them to specify the mesh size.

Figure 7.2 shows surfaces of revolutions with mesh sizes of 4 x 4 and 20 x 20. It should be mentioned that a finer mesh size for a surface does not improve its mathematical representa­tion; it only improves its visualization. Finally, some CAD/CAM systems do not permit their users to delete curves used to create surfaces unless the latter are deleted first.

• Following are descriptions of major surfaces: 1. Plane surface: It is the simplest surface. It requires three non­coincident points to define an infinite plane. The plane surface can be used to generate cross sections by intersecting a solid with it. Figure 1 shows planar surfaces. 2. Ruled (lofted) surface: It is a linear surface. It interpolates linearly between two boundary curves that define the surface (rails). Rails can be any curves. This surface is ideal for representing surfaces that do not have any twists or kinks. Figure 2 gives some examples.

Figure 1. Plane surface

Figure 2. Ruled surface

3. Surface of revolution: It is an axisymmetric surface that can model axisymmetric objects. It is generated by rotating a planar curve in space about the axis of symmetry a certain angle as shown in Figure 3. 4. Tabulated cylinder: It is a surface generated by translating a planar curve a certain distance along a specified direction (axis of the cylinder or directrix) as shown in Figure 4. The plane of the curve is perpendicular to the directrix. This surface is not lit­erally a cylinder. It is used to generate extruded surfaces that have identical cross sections.

Figure 3. Surface of revolution

Figure 4. Tabulated cylinder

• 5. Bezier surface: It is a surface that approximates or interpolates given input data. It is different from the previous surfaces in that it is a synthetic surface. It extends the Bezier curve to surfaces. It is a general surface that permits twists, and kinks. Bezier surface allows only global control of the surface. Figure 5 shows a Bezier surface.

Figure 5. Bezier surface

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6. B­spIine surface: It is a surface that can approximate or interpolate given input data. Figure 7.8 shows an interpolating example. It is a synthetic surface. It is a general surface like a Bezier surface but with the advantage of permitting local control of the surface. • 7. Coons surface: The previously described surfaces are used with either open boundaries or given data points. A Coons patch is used to create a surface using curves that form closed boundaries as shown in Figure 7.9.

8. Fillet surface: It is a B­spline surface that blends two surfaces together as shown in Figure 7.10. The two original surfaces may or may not be trimmed. 9. Offset surface: Existing surfaces can be offset to create new ones identical in shape but with different dimensions. It is a useful surface to use to speed up surface creation. For example, to create a hollow cylinder, the outer or inner cylinder can be created using a cylinder command and the other one can be created by an offset command. The offset surface command becomes very efficient to use if the original surface is a composite one. Figure 7.11 shows an offset surface.

Figure 7.10 Fillet surface.

Figure 7.10 offset surface.

Example

Solid

Solid models are known to be complete, valid, and unambiguous representations of objects. Simply stated, a complete solid is one which enables a point in space to be classified relative to the object, if it is inside, outside, or on the object. This classification is sometimes call spatial addressability. A valid solid is one that does not have dangling edges or faces. An unambiguous solid has one and only one interpretation. Solid modeling achieves completeness, validity, and unambiguity of geometric models.

CAD systems offer two approaches to creating solid models: primitives and features. The former approach allows designers to use predefined shapes (primitives) as building blocks to create complex solids. Designers must use Boolean operations to combine the primitives. This approach is limited by the restricted shapes of the primitives. The features are more flexible as they allow the construction of more complex and elaborate solids than what the primitives offer. Some CAD systems offer both approaches, while others offer only the features approach.

• Consider the object shown in Figure 9.1 to illustrate the two approaches. We can create a block and subtract six cylinders from it using the primitives approach. Or, we can create a rectangle with six circles inside it in the Top sketch plane and extrude it using the features approach. The resulting solid is the feature in this case. Figure 9.1 A typical solid model.

Geometry and Topology • A solid model of an object consists of both the topological and geometrical data of the object. The completeness and unambiguity of a solid model are attributed to the fact that its database stores both its geometry and its topology. The difference between geometry and topology is illustrated in Figure 9.2. Geometry (sometimes called metric information) is the actual dimensions that define the entities of the object. The geometry that defines the object shown in Figure 9.2 is the lengths of lines L1, L2, and L3, the angles between the lines, and the radius R and the center P1 of the half circle.

Topology (sometimes called combinatorial structure) is the connectivity and associativity of the object entities. It has to do with the notion of neighborhood; that is, it determines the relational information between object entities. The topology of the object shown in Figure 9.2b can be stated as follows: L1 shares a vertex (point) with L2 and C1, L2 shares a vertex with L1, and L3, L3 shares a vertex with L2 and C1, L1 and L3 do not overlap, and P1 lies outside the object. Based on these definitions, neither geometry nor topology alone can completely define objects.

While solid models are complete and unambiguous, they are not unique. An object may he constructed in various ways. Consider the object shown in Figure 9.3. Using the primitive approach, one can construct the solid model of the object by dividing it into two blocks and a cylinder. We can add the two blocks first and then subtract the cylinder (Figure 9.36), or we can subtract the cylinder from a block and add the other block to the resulting subsolid (Figure 9.3c). Figure 9.4 shows two alternatives (create different cross sections and extrude them) if we use the features approach. Regardless of the order and method of construction, the resulting solid model of the object is always complete and unambiguous. However, there will always be one way that is more efficient than others to construct solid models, as is the case with curves and surfaces.

General types of solid

More explanation on solid

Solid Entities The entities we use to create solid models depend on the approach we use. The primitives approach uses primitives and the features approach uses sketches. Many CAD systems provide both approaches to increase their modeling domain. Let look at the basics of primitives.

Primitives are considered building blocks. Primitives are simple, basic shapes which can be combined by a mathematical set of Boolean operations to create the solid. Primitives themselves are considered valid off­the­shelf solids. The user usually positions primitives as required before applying Boolean operations to construct the final solid.

There is a wide variety of primitives available commercially to users. However, the four most commonly used ones are the block, cylinder, cone, and sphere. These are based on the four natural quadrics: planes, cylinders, cones, and spheres. For example, the block is formed by intersecting six planes. These quadrics are considered natural because they represent the most commonly occurring surfaces in mechanical design which can be produced by rolling, turning, milling, cutting, drilling, and other machining operations used in industry.

Following are descriptions of the most commonly used primitives •

• • • • •

1. Block. This is a box or cube whose geometrical data is its width, height, and depth. Its local coordinate system XL,YL,ZL is shown in Figure 1. Point P defines the origin of the XLYL,ZL system. The signs of W, H, and D determine the position of the block relative to the coordinate system. For example, a block with a negative value of W is displayed as if de­block shown in Figure 9.5 is mirrored about the XL,ZL, plane. 2. Cylinder. This primitive is a right circular cylinder whose geometry is defined by its radius R (or diameter D) and length H. The length H is usually taken along the direction the ZL axis. H can be positive or negative. 3. Cone. This is a right circular cone or a frustum of a right circular cone whose base diameter R, top diameter (for truncated cone), and height H are user­defined. 4. Sphere. This is defined by its radius R or diameter D and is centered about the origin of its local coordinate system. 5. Wedge. This is a right angled wedge whose height H. width W, and base depth D form its geometric data. 6. Torus. This primitive is generated by the revolution of a circle about an axis lying in its plane ZL axis in Figure 1. The torus geometry can be defined by the radius (or diameter) of its body R1 and the radius (or diameter) of the centerline of the torus body R2, or the geometry can be defined by the inner radius (or diameter) R1 and outer radius R0.

• All these primitives can be created using the features approach. They are all 21/2 D objects. The block, cylinder, and wedge are uniform thickness. The cone, sphere, and torus are axisymmetric. This explains why some CAD systems such as Pro/E,SolidWorks and CATIA do not offer them — the user can generate them via sketching. This simplifies software development as there is no need to write separate primitives' functions.

Figure 1. Most common primitives

Two or more primitives can be combined to form a solid. To ensure the validity of the resulting solid, the allowed combinatorial relationships between primitives are achieved via Boolean (or set) operations. The available Boolean operators are union (U or +), intersection (n or I), and difference ( ­ ). The union operator is used to combine or add together two objects or primitives. Intersecting two primitives gives a shape equal to their common value. The difference operator is used to subtract one object from the other and results in a shape equal to the difference in their volumes.

Figure 8. A typical solid and its building primitives

End of lecture