Development By Triangulation 5w5317
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DEVELOPMENT BY TRIANGULATION
TRIANGULAR DEVELOPMENT Triangulation is slower and more difficult than parallel line or radial line development, but it is more practical for many types of figures. Additionally, it is the only method by which the developments of warped surfaces may be estimated. In development by triangulation, the piece is divided into a series of triangles as in radial Line development. However, there is no one single apex for the triangles. The problem becomes one of finding the true lengths of the varying oblique lines. This is usually done by drawing a true, length diagram.
A BRIEF LOOK AT PARALLEL LINE DEVELOPMENT
THE VIEW ON THE LEFT SHOWS THE DEVELOPMENT OF A TRUNCATED CYLINDER. PARALLEL LINES
A BRIEF LOOK AT RADIAL LINE DEVELOPMENT
RADIAL LINES OF A CONE
Triangulation Development In this method of development the surface of the object is divided into a number of triangles. The true sizes of the triangles are found and they are drawn in order, side by side, to produce the pattern. It will be apparent that to find the true sizes of the triangles it is first necessary to find the true lengths of their sides.
TRUE LENTHS
1. REBATEMENT OR ROTATION METHOD 2. TREE METHOD
EXAMPLE 1 REBATEMENT METHOD c
d o
Before starting with the development you must find the true lengths of sides ‘oa’, ‘ob’, ‘oc’ and ‘od’. The base lines, lines ‘ab’, ‘bc’, ‘cd’ and ‘da’
HP
a
VP
PLAN
b
o
ad
bc ELEVATION
are all true lengths since they are all parallel or perpendicular to the reference line HP/VP.
c
d o
a
PLAN
b
o
TL
a1
ad
bc ELEVATION
The rotation method is used to find the true length of line ‘oa’.
c
d o
a
True lengths of the other lines are worked out.
b
PLAN
Notice how the drawing is becoming cluttered with lines.
o
TL
TL TL TL
a1
d1 ad
bc ELEVATION
c1
b1
c
d o
To draw the development, you start by drawing one triangle first. a
In the example, triangle ‘oab’ is drawn first.
b
PLAN
Draw second triangle, triangle ‘obc’.
o
TL
Draw third triangle, triangle ‘ocd’.
TL TL
Draw last triangle, triangle ‘oda’.
TL a1
d1 ad
bc
c1
b1
Do not forget to use the true lengths of the lines.
ELEVATION
a
a o
d
b
c DEVELOPMENT
EXAMPLE 1 TRUE LENGTH TREE METHOD
Develop the given oblique cone.
O
EXAMPLE 2 ROTATION METHOD O
PLAN
TRUE LENGTHS
PLAN
7
6,8
5,9 9
4,10
3,11
10
1
1
2,12
3,11
4,10
5,9
7
11 1
8
ELEVATION
Develop the given oblique cone.
O
7
1
Divide the oblique cone into a number of triangles.
6 5
3
4 ELEVATION
The most convenient number is twelve since you can easily divide the base into twelve divisions and the divisions to the apex. O
1
12
11
10
Use the rotation method to find the true lengths of all lines.
Using the true lengths of the sides, draw the triangles one at a time.
2 9
3 4
8 5
6
7
DEVELOPMENT
Do not forge to start from the shortest side.
Method 2 TRUE LENGTH TREE METHOD O
O
PLAN
7
6,8
5,9 9
TRUE LENGTHS
4,10 10
3,11
1
1
2,12
3,11
4,10
TRUE LENGTH TREE
11
8
7
1
O
6 5 4
O
3
ELEVATION
1
12
11
10 2 9
3 4
8 5
6
7
5,9
7
TRUNCATED OBLIQUE CONE
Transition Piece Often in industry it is necessary to connect tubes and ducts of different cross-sectional shapes and areas, especially in air conditioning, ventilation and fume extraction applications. The required change in shape and 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 area; for example square-to-round.
EXAMPLE 1 Develop the given square to square transition piece. 1
c
d
a
3 4 4
2 1
a
3 b
PLAN 4
1,3
1
2
TL
a1,d1
d
2
TL
ad
bc
c
b1,c1
DEVELOPMENT a
b
ELEVATION
The rotation method is used to find the lengths of the sides.
1
c
d
a
3 4 4
2 1
a
3 b
PLAN 4
1,3
d
2 1,2,3,4
1
2
TL
ad
bc ELEVATION
c
DEVELOPMENT a,b,c,d
a
b
TL TREE
The true length tree is used to find the true lengths of the sides.
EXAMPLE 2 Develop the given circle to rectangle transition piece.
a
c
d
d 1 10
7
8
4 5
9
10
7
a
12 1
1,7
2
b
PLAN 10
11
5
4
4
7
6
5
4
3
c
TL TL
a,d
b,c ELEVATION
a
DEVELOPMENT
b
The rotation method is used to find the true lengths of the lines.
a
c
d
d 1 10
7
8
4 5
9
10
7
a
12 1
1,7
2
b
PLAN 10
11
TL 2,3,5,6,8,9,11,12 4
7
6
5
4
3
c TL 4,1,10,7
a,d
b,c ELEVATION
TL TREE
a
DEVELOPMENT
The true length tree is used to find the true lengths.
b
EXAMPLE 3 a
1 4
1
a
g 4 3
2
1
f
3 2
g
e b
PLAN 2,1
d
c
f
3,4
e d c
a b
DEVELOPMENT
a,b
e ELEVATION
d
e f,g
b a,b
b
a
c d
b
TRUE LENGTH TREE
The true length tree is used to find the true lengths.
EXAMPLE 4 CIRCLE TO SQUARE
EXAMPLE 5 CIRCLE TO RECTANGLE
TRIANGULAR DEVELOPMENT Triangulation is slower and more difficult than parallel line or radial line development, but it is more practical for many types of figures. Additionally, it is the only method by which the developments of warped surfaces may be estimated. In development by triangulation, the piece is divided into a series of triangles as in radial Line development. However, there is no one single apex for the triangles. The problem becomes one of finding the true lengths of the varying oblique lines. This is usually done by drawing a true, length diagram.
A BRIEF LOOK AT PARALLEL LINE DEVELOPMENT
THE VIEW ON THE LEFT SHOWS THE DEVELOPMENT OF A TRUNCATED CYLINDER. PARALLEL LINES
A BRIEF LOOK AT RADIAL LINE DEVELOPMENT
RADIAL LINES OF A CONE
Triangulation Development In this method of development the surface of the object is divided into a number of triangles. The true sizes of the triangles are found and they are drawn in order, side by side, to produce the pattern. It will be apparent that to find the true sizes of the triangles it is first necessary to find the true lengths of their sides.
TRUE LENTHS
1. REBATEMENT OR ROTATION METHOD 2. TREE METHOD
EXAMPLE 1 REBATEMENT METHOD c
d o
Before starting with the development you must find the true lengths of sides ‘oa’, ‘ob’, ‘oc’ and ‘od’. The base lines, lines ‘ab’, ‘bc’, ‘cd’ and ‘da’
HP
a
VP
PLAN
b
o
ad
bc ELEVATION
are all true lengths since they are all parallel or perpendicular to the reference line HP/VP.
c
d o
a
PLAN
b
o
TL
a1
ad
bc ELEVATION
The rotation method is used to find the true length of line ‘oa’.
c
d o
a
True lengths of the other lines are worked out.
b
PLAN
Notice how the drawing is becoming cluttered with lines.
o
TL
TL TL TL
a1
d1 ad
bc ELEVATION
c1
b1
c
d o
To draw the development, you start by drawing one triangle first. a
In the example, triangle ‘oab’ is drawn first.
b
PLAN
Draw second triangle, triangle ‘obc’.
o
TL
Draw third triangle, triangle ‘ocd’.
TL TL
Draw last triangle, triangle ‘oda’.
TL a1
d1 ad
bc
c1
b1
Do not forget to use the true lengths of the lines.
ELEVATION
a
a o
d
b
c DEVELOPMENT
EXAMPLE 1 TRUE LENGTH TREE METHOD
Develop the given oblique cone.
O
EXAMPLE 2 ROTATION METHOD O
PLAN
TRUE LENGTHS
PLAN
7
6,8
5,9 9
4,10
3,11
10
1
1
2,12
3,11
4,10
5,9
7
11 1
8
ELEVATION
Develop the given oblique cone.
O
7
1
Divide the oblique cone into a number of triangles.
6 5
3
4 ELEVATION
The most convenient number is twelve since you can easily divide the base into twelve divisions and the divisions to the apex. O
1
12
11
10
Use the rotation method to find the true lengths of all lines.
Using the true lengths of the sides, draw the triangles one at a time.
2 9
3 4
8 5
6
7
DEVELOPMENT
Do not forge to start from the shortest side.
Method 2 TRUE LENGTH TREE METHOD O
O
PLAN
7
6,8
5,9 9
TRUE LENGTHS
4,10 10
3,11
1
1
2,12
3,11
4,10
TRUE LENGTH TREE
11
8
7
1
O
6 5 4
O
3
ELEVATION
1
12
11
10 2 9
3 4
8 5
6
7
5,9
7
TRUNCATED OBLIQUE CONE
Transition Piece Often in industry it is necessary to connect tubes and ducts of different cross-sectional shapes and areas, especially in air conditioning, ventilation and fume extraction applications. The required change in shape and 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 area; for example square-to-round.
EXAMPLE 1 Develop the given square to square transition piece. 1
c
d
a
3 4 4
2 1
a
3 b
PLAN 4
1,3
1
2
TL
a1,d1
d
2
TL
ad
bc
c
b1,c1
DEVELOPMENT a
b
ELEVATION
The rotation method is used to find the lengths of the sides.
1
c
d
a
3 4 4
2 1
a
3 b
PLAN 4
1,3
d
2 1,2,3,4
1
2
TL
ad
bc ELEVATION
c
DEVELOPMENT a,b,c,d
a
b
TL TREE
The true length tree is used to find the true lengths of the sides.
EXAMPLE 2 Develop the given circle to rectangle transition piece.
a
c
d
d 1 10
7
8
4 5
9
10
7
a
12 1
1,7
2
b
PLAN 10
11
5
4
4
7
6
5
4
3
c
TL TL
a,d
b,c ELEVATION
a
DEVELOPMENT
b
The rotation method is used to find the true lengths of the lines.
a
c
d
d 1 10
7
8
4 5
9
10
7
a
12 1
1,7
2
b
PLAN 10
11
TL 2,3,5,6,8,9,11,12 4
7
6
5
4
3
c TL 4,1,10,7
a,d
b,c ELEVATION
TL TREE
a
DEVELOPMENT
The true length tree is used to find the true lengths.
b
EXAMPLE 3 a
1 4
1
a
g 4 3
2
1
f
3 2
g
e b
PLAN 2,1
d
c
f
3,4
e d c
a b
DEVELOPMENT
a,b
e ELEVATION
d
e f,g
b a,b
b
a
c d
b
TRUE LENGTH TREE
The true length tree is used to find the true lengths.
EXAMPLE 4 CIRCLE TO SQUARE
EXAMPLE 5 CIRCLE TO RECTANGLE
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