MEL 110

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MEL 110

• Development of surfaces

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Development is a graphical method of obtaining
the area of the surfaces of a solid. When a solid
is opened out and its complete surface is laid on
a plane, the surface of the solid is said to be
developed. The figure thus obtained is called a
development of the surfaces of the solid or
simply development. Development of the solid,
when folded or rolled, gives the solid.
Examples
Prism Made up of same number of rectangles as
sides of the base One side Height of the
prism Other side Side of the base Cylinder
Rectangle One side Circumference of the
base Other side Height of the cylinder Pyramid
Number of triangles in contact The base may be
included if present
h
pd
fd
T. L.
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Methods used to develop surfaces

• Parallel-line development Used for prisms,
  cylinders etc. in which parallel lines are drawn
  along the surface and transferred to the
  development.
• Radial-line development Used for pyramids, cones
  etc. in which the true length of the slant edge
  or generator is used as radius.
• Triangulation development Complex shapes are
  divided into a number of triangles and
  transferred into the development (usually used
  for transition pieces).
• Approximate method Surface is divided into parts
  and developed. Used for surfaces such as spheres,
  paraboloids, ellipsoids etc.
• Note- The surface is preferably cut at the
  location where the edge will be smallest such
  that welding or other joining procedures will be
  minimal.

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Parallel line development This method is
employed to develop the surfaces of prisms and
cylinders. Two parallel lines (called stretch-out
lines) are drawn from the two ends of the solids
and the lateral faces are located between these
lines.
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Development of lateral surfaces of different
solids. (Lateral surface is the surface excluding
top base)
Cylinder A Rectangle
Pyramids (No.of triangles)
Cone (Sector of circle)
L
?
H Height D base diameter
Prisms No.of Rectangles
L Slant edge. S Edge of base
Radial-line development
Parallel-line development
H Height S Edge of base
Cube Six Squares.
Tetrahedron Four Equilateral Triangles
All sides equal in length
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DEVELOPMENT OF FRUSTUM OF CONE
DEVELOPMENT OF FRUSTUM OF SQUARE PYRAMID
Base side
Top side
?
R Base circle radius of cone L Slant height of
cone L1 Slant height of cut part.
L Slant edge of pyramid L1 Slant edge of cut
part.
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Cube cut by section plane
Project, horizontally, the points of intersection
of the cutting plane with the edges. Mark
distances 3M, 3N
4, d
2, b
4
1
1
3
2
D
A
C
B
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Draw the development of the lower portion of the
cone surface cut by a plane. Cone base diameter
is 40 mm and height is 50 mm. The cutting plane
intersects the cone axis at an angle of 45o and
20 mm below the vertex

• Divide the cone in the top view and project the
  corresponding generator lines in the front view
• Develop the complete surface of the cone by
  drawing an arc with radius length of side
  generator of cone and length of arc
  circumference of cone base
• Draw the corresponding generator lines
• Obtain true lengths of o1, o2 etc. by auxiliary
  view, rotation method OR by projecting onto one
  of the side generators (which are in true length)
• Mark the distances (true lengths) o1, o2etc. in
  the development and join them to get the
  development of the lower portion of the cone

Radius of cone R
True lengths b2, 2o obtained by auxiliary view
method
o
a
g
2
b
f
c
e
d
o
T
F
o
4
1
3
2
4
2
3
l
a
a
2
b
1
b
c
d
e
f
g
True length of (o2, o3) (o2, o3) etc.
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If R 2r then ? 180, i.e., if the slant
height of a cone is equal to its diameter of base
then its development is a semicircle of radius
equal to the slant height.
10
Develop the surface of the symmetrical half of an
oblique pyramid with a horizontal regular
hexagonal base (side 20 mm and vertex 30 mm above
one corner of the base)
Obtain true lengths of the edges ob and oc by
rotation or auxiliary view method Edge oa is seen
in true length in the Front View ab bc cd
side of hexagonal base 20 mm
od and dc can be constructed as they are
perpendicular to each other The lengths of bc,
and ob are known and therefore these distances
can be marked with the compass After drawing
triangles odc and ocb, triangle oba can be
completed
o, d
b
a
c
T
c
b
o
F
o
a
b
c
d
c
c
a
b
d
b
a
True lengths
d
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Develop the surface of the cylinder which is cut
as shown

• Divide the base of the cylinder in the top and
  front views into the a certain number of equal
  parts (12 here)
• Develop the surface of the cylinder (rectangle
  with length p x diameter and height height of
  cylinder) and divide it into the same number of
  equal parts
• The projector lines from the top view intersect
  the cut portion of the cylinder at a, b, c..f.
• Project these points onto the developed surface

f50
T
F
g
g
f
h
i
e
45o
f,h
e,i
j
d
d,j
k
c,k
c
l
b,l
a
a
b
15o
a
100
30o
h
i
px50
h
h
12
Oblique square prism
e, j, h
f, l, g
a, i, d
b, k, c
a, e
b, f
i, j
k, l
d, h
c, g
13
Oblique prism
d
e
c
f
g
h
a
b
i
a
b
f
h
i
g