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Intersections and developments are commonly found in many engineering disciplines. Two surfaces that meet to form a line of intersection. A pipe going through a wall is another example of an intersection. As an engineer you must be able to determine the exact point of intersection between the pipe and the wall. A development is the outside surface of a geometric form laid flat. Sheet metal products are an application for developments. A cardboard cereal box, or the heating and air conditioning ductwork used in buildings and large aircraft, when laid flat are examples of developments. As an engineer you should be able to create the development of common shapes, such as cones, prisms, and pyramids.
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INTERSECTIONS AND DEVELOPMENTS |
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| 13.1 The oil and chemical industries use intersections when they design and build refineries. |
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| 13.2 Developments have and still continue to be used in the ducts of heating and cooling systems. |
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| 13.3 The aircraft industry uses developments in the design and production of automobiles and aircraft. |
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INTERSECTIONS |
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Visibility is the clear and correct representation of the relative positions or two geometric figures, in multiview drawings. |
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| 13.4 The basic concept of visibility can be demonstrated with two skew lines, in which one line is either in front of, behind, above, below, to the left, or to the right of the other line. |
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| 13.5 This same procedure can be used to determine the visibility of two cylinders, such as plumbing pipes or electrical conduct. |
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| 13.6 Using adjacent views to determine visibility of a tetrahedron. |
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| 13.7 An intersection is a point or line where two geometric forms, such as lines or surfaces, meet or cross each other. Two lines that intersect share a common point. If the lines do not have a common point that projects from view to view, the lines are nonintersecting. The intersection of a line and a plane is referred to as the piercing point. A line will intersect a plane if the line is not parallel to the plane. |
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| 13.8 In the edge view method, the plane is shown as an edge, which means that all points on the plane can be represented on that edge. |
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| 13.9-10 The cutting plane method can also be used to find the intersection between a plane and a line. |
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| 13.11 The intersection of two planes is a straight line all of whose points are common to both planes. The line of intersection between two planes is determined by locating the piercing points of lines from one plane with the other plane and drawing a line between the points. |
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| 13.12 Using the cutting plane method to determine the intersection of two planes. |
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| 13.13 To find the intersection between a plane and a solid, determine the piercing points using either cutting planes or auxiliary views, then draw the lines of intersection. |
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| 13.14 When an oblique plane intersects a prism, an auxiliary view can be created to show an edge view of the oblique plane. |
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| 13.15 The line of intersection between an oblique plane and a cylinder is an ellipse, which can be formed by creating an auxiliary view that shows the plane as an edge. |
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| 13.16 When a plane intersects a cone and does not intersect the base, the line of intersection can be found by passing multiple cutting planes through the cone, either parallel or perpendicular to the base. |
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| 13.17 Multiple cutting planes can also be sued to determine the intersection between an oblique plane and a right circular cone. |
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| 13.18 The intersection between two prisms is constructed by developing an auxiliary view in which the faces of one of the prisms are shown as an edge. Piercing points between the edges and the projection of the second prism are located and are projected back into the principal views. |
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| 13.19 Straight lines of intersection are created when a prism intersects a pyramid. A combination of the cutting plane and edge view methods can be used to create this line of intersection. |
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| 13.20 The intersection between a cone and a prism is a curved line, which is determined using an auxiliary view and cutting planes. |
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| 13.21 To determine the line of intersection between a prism and a cylinder, use either an auxiliary view showing the intersecting sides of the prism on edge, or use cutting planes. |
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| 13.22 The intersection between a cylinder and a cone is a curved line. Cutting planes are used to locate the intersection points. |
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| 13.23 The intersection between two perpendicular cylinders produces a curved line of intersection. Cutting planes are used to locate the intersection points. |
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| 13.24 The line of intersection between two nonperpendicular cylinders is a curved line and is constructed using cutting planes. |
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| 13.25 With 3-D CAD, lines of intersection can be determined by Boolean-based operations. |
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DEVELOPMENTS |
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| 13.26A A development is the unfolded or unrolled, flat or plane figure of a 3-D object. Called a pattern, the plane figure may show the true size of each area of the object. When the pattern is cut, it can be rolled or folded back into the original object. |
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| 13.26B A true development is one in which no stretching or distortion of the surface occurs, and every surface of the development is the same size and shape as the corresponding surface on the 3-D object. Only polyhedrons and single-curved surfaces can produce true developments. |
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| 13.26C Polyhedrons are composed entirely of plane surfaces that can be flattened true size onto a plane in a connected sequence. |
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| 13.26D Single-curved surfaces are composed of consecutive pairs of straight-line elements in the same plane. |
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| 13.26E An approximate development is one in which stretching or distortion occurs in the process of creating the development. The resulting flat surfaces are not the same size and shape as the corresponding surfaces on the 3-D object. Warped surfaces do not produce true developments, because pairs of consecutive straight-line elements do not form a plane. Double-curved surfaces such as spheres do not produce true developments. |
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| 13.26A-B Parallel line developments are made from common solids that are composed of parallel lateral edges or elements. Prisms and cylinders are solids that can be flattened or unrolled into a flat pattern, and all parallel lateral surfaces or elements will retain their parallelism. |
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| 13.26C-D Radial line developments are made from figures such as cones and pyramids. In the development, all the elements of the figure become radial lines that have the vertex as their origin. |
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| 13.26E Approximate developments are used for double-curved surfaces, such as spheres. Approximate developments are constructed through the use of conical sections of the object. |
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| 13.26F Triangulation developments are made from polyhedrons, single-curved surfaces, and warped surfaces. These developments involve sub dividing any ruled surface into a series of triangular areas. Ployhedron triangulation results in a true development. Triangulation for single-curved surfaces increases in accuracy through the use of smaller and more numerous triangles. Triangulation developments of warped surfaces produce only approximations of those surfaces. Developments of objects with parallel elements or parallel lateral edges are begun by constructing a stretch-out line that is parallel to a right section of the object and is therefore perpendicular to the elements or lateral edges. The distances around the object is then laid out along the stretch-line. |
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| 13.27 Typically, developments are constructed using an inside pattern. This allows bending machines or handbent materials to hide any markings on the inside of the object. |
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| 13.28 The fold lines of the pattern are represented as either thin solid lines or dashed lines and are the seam where the material will be bent. |
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| 13.29 In the development of a right rectangular prism all lateral edges in the front view are parallel to each other and are true length. The lateral edges are also true length in the development. The length, or the stretch-out line, of the development is equal to the true distance around a right section of the object. |
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| 13.30 The same basic steps are used to create a truncated prism development as were described for the rectangular prism. |
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| 13.31 A right circular cylinder is developed in a manner similar to that used for a right rectangular prism. |
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| 13.32 A truncated cylinder is developed similar to a cylinder. |
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| 13.33 A right circular cone is developed using the radial line method. |
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| 13.34 A truncated cone is developed by transferring radial distances. |
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| 13.35 An oblique cone triangulation method is used to construct the development of oblique cones. |
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| 13.36 A truncated pyramid with no shown true length edges is developed by the triangulation method. |
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| 13.37 To construct the development of an oblique prism, create a right section of the prism and a true-size auxiliary view. |
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| 13.38 The development of an oblique cylinder is similar to that of an oblique prism with an infinite number of sides. A right section of the cylinder is created and a true-size auxiliary view is constructed. |
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| 13.39 Transition pieces are used to connect two differently sized or shaped openings, or two openings in skewed positions, such as those found in heating and ventilation systems. To create the development of a transition piece, first determine the geometric shape of the piece. |
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| 13.40 The development of a transition piece between a regular prism and a cylinder. |
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| 13.41A Warped and double-curved surfaces can only be developed by approximation. An example of a double-curved surface is a sphere. The surface is developed by dividing it into a series of zones. Each zone is then approximated as a truncated right circular cone, which is developable. This is called the polyconic method. To create the development cut the sphere into zones by passing multiple cutting planes through the sphere perpendicular to the axis. Each section thus formed becomes an approximation of a truncated cone, with side elements that are curved instead of straight. |
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| 13.41B The polycylindric method involves the development of a sphere by passing multiple cutting planes parallel to and through the axis, thus dividing the surface into equal sections. Each section of the sphere is developed as a cylindrical surface. |
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| 13.42 Using CAD techniques to develop surfaces of objects. |
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SUMMARY |
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Determining the intersection between surfaces is a systematic procedure. Most intersections can be determined using auxiliary views, cutting planes, or a combination of both to determine a number of piercing points. These piercing points are then connected with lines or curves depending on the types of surfaces that are intersecting. Visibility of the lines of intersection is then determined using projection methods. Developments of surfaces also use auxiliary views and cutting plane to determine the fattened shape various geometric forms, such as cones, cylinders, and pyramids. Developments are either parallel line, radial line, triangulation, or approximation. Parallel and radial line and triangulation can result in true development of surfaces. However, some surfaces such as warped and double curved can only be approximated using triangulation or approximate methods. |