Development of Surfaces Chapter Eleven Intersections and Development of Surfaces

Purpose This chapter provides an overview of how to: construct parallel line developments of prisms and cylinders construct radial line developments of pyramids and cones draw the line of intersection between geometric surfaces and construct their development construct the development of transition pieces, using triangulation.

Development of prisms Figure 11.1(a), p.292, shows a pictorial view of a rectangular right prism with open ends. This prism consists of four rectangular sides which, when folded out on to a flat surface, form the area necessary to make the prism. This area is called the development of the prism or the pattern for the prism. Figure 11.1(b) is a view showing the prism unfolding on to a flat surface, while Figure 11.1(c) is the complete layout of the surface of the prism when it is unfolded.

Development of prisms Figure 11.2, p.292, illustrates the development of a truncated right prism shown on the left of the figure. To obtain the development, follow the steps 1–6. A practical application of a truncated prism is shown in Figure 11.3, which illustrates an elbow in rectangular pipe. The development of one half of the elbow is shown on the right.

Development of prisms Figure 11.4(a), p.293, is a pictorial view of a hexagonal right prism with open ends. This prism consists of six rectangular sides. Figures 11.4(b) and (c) illustrate how the development of this prism is obtained. Figure 11.5 shows the development of a truncated hexagonal right prism. Figure 11.6 shows the development of a truncated oblique hexagonal prism. To obtain it, follow steps 1–5.

Development of prisms Square, pentagonal and octagonal, right and oblique prisms are developed in a similar manner. Problems 11.11 and 11.12 (p.320) are two lobster- back bends made up of truncated square and hexagonal prisms respectively, called segments. If it is not desirable to have a reduction in cross- sectional area of the bend, the segments must be designed and fitted according to Problem 11.11. More segments may be inserted in the bend than shown, in order to make the change of direction smoother and to approximate a radial bend.

Line of intersection—cylinders and cones The line of intersection of two or more intersecting surfaces has to be determined in order to develop any of the surfaces. Element method – this involves the use of line elements drawn on the surfaces of the intersecting shapes and passing through the area where the line of intersection occurs. For example, cone and cylinder intersection (Figure 11.7, p.294, steps 1–5).

Line of intersection—cylinders and cones Cutting plane method – this involves drawing a series of horizontal cutting planes, each of which cuts through both the intersecting surfaces, for example a cone (to give a circle) and a cylinder (to give a rectangle). For example, cone and cylinder intersection (Figure 11.8, p.295, steps 1–7).

Line of intersection—cylinders and cones Common sphere method – when intersecting cylinders and cones envelop a common sphere, the line(s) of intersection are straight when viewed from the side. See Figure 11.9, p.296. For example, cone and cylinder intersection (Fig. 11.9(a), steps 1 and 2). For example, cone and two cylinders intersection (Fig. 11.9(b), steps 1–5).

Development of cylinders A right cylinder is a closed circular surface. The shape of the cross-section of all right cylinders at right angles to the axis is circular. The development of a right cylinder is illustrated in Figure 11.10, p.296, where a series of pictorial views (a), (b) and (c) show how the cylinder can be unrolled from the formed position at (a) to the flat rectangular surface at (c), the dimensions of which are equal to the circumference and length of the cylinder.

Development of cylinders If truncations and/or intersections must be plotted on the development, this length should be divided geometrically into twelve equal parts to create the surface element lines. The method described at 5.15, p.108, is ideal for dividing the circumference into twelve equal parts. A truncated right cylinder is one which is cut at an angle to the axis. A practical example of its use is in the construction of the cylindrical elbow shown in Figure 11.11(a), p.297. This is made up of two truncated right cylinders joined together along the axis of truncation to form the included angle of the bend, in this case 120°.

Development of cylinders A truncated right cylinder is obtained by following the steps 1–5, p.296. An oblique cylinder can be defined as a closed curved surface in which the shape of the cross- section at right angles to the axis is elliptical. One particular cross-section at an angle to the axis is circular, and it is at this cross-section that the joining of right cylindrical pipes takes place. See Figure 11.12(a), p.297. To obtain the development, follow the steps 1–7 (refer to Figure 11.12(b))

Elbows Elbows are used to change the direction of pipes in round, square, hexagonal and other cross-sections and are normally made of two, three, four or more pieces. The number of pieces depends on the cross-sectional shape and area. To develop the four-piece round elbow shown in Figure 11.13, follow the steps 1–6, p.298.

Development of T pieces The development of cylindrical T pieces involves finding the line of intersection of the two cylinders, and then drawing the development as shown in Figure 11.11. To develop both branches of the oblique T piece, follow the steps 1–7, p.299, and refer to Figure 11.14. To develop both branches of the offset oblique T piece, refer to Figure 11.15 and follow the steps 1–7, p.300.

Development of T pieces Figure 11.16 (p.302) illustrates the development of an oblique cylindrical connecting pipe with a cylindrical pipe insert. The development of the connecting pipe without the hole for the insert is described on page 297. One method to find the line of intersection between the insert and the connecting pipe for this problem are the steps 1–8, p.301. Figure 11.17 (p.303) steps 1–3 illustrates a second method of obtaining the line of intersection between the insert and the connecting pipe.

Development of pyramids A right pyramid may be defined as a surface with a number of identical triangular sides which have a common apex situated vertically above the centre of the base. An important fact to remember about all right pyramids is that the sloping edges may be totally contained within the surface of an enveloping cone. This is illustrated in Figure 11.18(a) on page 303.

Development of pyramids Figure 11.18(b) on p.303 is a pictorial view of a hexagon-based right pyramid – its developed by following the steps 1–6, p.301. The oblique pyramid (Figure 11.19, p.304) may be defined as a surface with a number of flat unequal triangular sides which have a common apex not situated vertically above the centre of the base. Refer to Figure 11.19 as you read through the steps 1–10, p.304.

Development of cones A right cone can be defined as a surface which has a circular base and a curved sloping side which radiates from a point situated vertically above the centre of the base. This point is called the apex of the cone. The length of any straight line drawn down the sloping side from the apex to the base is constant and is called the slant height of the cone. Refer to Figure 11.20(c), p.305, for the development of a right cone which is obtained by following steps 1 and 2, p.306.

Development of cones Right cone truncated parallel to the base, refer to Figure 11.21 and steps 1 and 2, p.306. Right cone truncated at an angle to the base, refer to Figure 11.22 and steps 1–5. Right cone-vertical cylinder intersection, refer to Figure 11.23 and steps 1–8.

Development of cones Truncated right cone-right cylinder intersection, refer to Figure 11.24 and steps 1–5, p.308. Right cone-right cylinder, oblique intersection, refer to Figure 11.25 and steps 1–7, p.309.

Oblique cone An oblique cone can be defined as a surface which has a circular base and a curved sloping side which radiates from a point not situated vertically above the centre of the base. The length of any straight line drawn down the sloping side from the apex to the base is not constant; hence, the oblique cone does not have a constant slant height, and its development is somewhat more complicated than that of the right cone.

Oblique cone–oblique cylinder intersection Oblique cone, refer to Figure 11.26 and steps 1–10, p.310. Oblique cone–oblique cylinder intersection, refer to Figure 11.27 and steps 1–7, p.311.

Development of breeches or Y pieces Development of breeches or Y pieces – a three-way junction between cylindrical pipes or between cylindrical pipes and conical sections. The angles between the various branches can be equal or unequal and each branch should envelop a common sphere represented on the front view by a circle. Once the front view has been drawn and the line of truncation determined, the development of the branches is merely that of truncated cylinders and cones. This is shown on each of the three exercises of Figures 11.28, 11.29 and 11.30 (p.312).

Development of breeches or Y pieces Breeches piece—equal angle, equal diameters; unequal angle, equal diameters, refer to both Figures 11.28 and 11.29. and steps 1 and 2, p.312. Breeches piece—cylinder and two cones, equal angle, refer to Figure 11.30 and steps 1 and 2.

Development of transition pieces Often in industry it is necessary to connect tubes and ducts of different cross-sectional shapes and areas, the required change in shape and/or area is achieved by developing a transition piece with an inlet of a certain shape and cross-sectional area, and an outlet of a different shape and/or area; for example square- to-round. The transition is achieved by a technique called triangulation which involves dividing the transition surface into a suitable number of triangular segments, finding the true shape of each and then laying these down side by side to form the true surface development.

Development of transition pieces Square-to-rectangle transition piece, refer to Figure 11.31 and steps 1–4, p.313. Round-to-round transition piece, refer to Figure 11.32 and steps 1–12, p.314. Square-to-round transition piece, refer to Figure 11.33 and steps 1–9, p.316.

Development of transition pieces Oblique hood, refer to Figure 11.34 and steps 1–6, p.316. Offset rectangle-to-rectangle transition piece, refer to Figure 11.35 and steps 1–5, p.317.

Summary To construct prisms, cylinders, pyramids and cones the objects need to be developed, that is flattened out to form a pattern for construction. In addition, when these items are intersected by other objects or shapes the lines of intersection and/or transitional pieces need to be carefully developed.