Chapter Eleven
Intersections and
Development of Surfaces
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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.
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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.
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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.
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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.
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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 lobsterback bends made up of truncated square and
hexagonal prisms respectively, called segments.
 If it is not desirable to have a reduction in crosssectional 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.
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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).
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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).
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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).
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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.
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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°.
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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 crosssection 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))
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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.
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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 squareto-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.
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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.
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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.
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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.
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