Development of Surfaces

Development of surfaces

A development is the unfold / unrolled flat / plane figure of a 3-D object. It is also called a pattern where the plane 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 as shown in figure 1.

Figure 1. Typical development of the surface of a cuboid.

Types of development
There are three major types of development followed by industries. Examples are shown in figure 2.

Parallel line development: In this parallel lines are used to construct the expanded pattern of each three-dimensional shape. The method divides the surface into a series of parallel lines to determine the shape of a pattern.
Radial line development: In this, lines radiating from a central point to construct the expanded pattern of each three-dimensional shape is used. These shapes each form part of a cone and lines radiating from the vertex of the cone generate the expanded pattern of the curved surface as shown in the following explorations.
Triangulation method: This is generally used for polyhedron, single curved surfaces, and warped surfaces.
Approximate development: In this, the shapes obtained are only approximate. After joining, the part is stretched or distorted to obtain the final shape

Figure 2. Typical examples of the various types of development.

A true development is one in which no stretching or distortion of the surfaces occurs and every surface of the development is the same size and shape as the corresponding surface on the 3-D object. e.g. polyhedrons and single curved surfaces.
As illustrated in figure 3, polyhedrons are composed entirely of plane surfaces that can be flattened true size onto a plane in a connected sequence, where as single curved surfaces are composed of consecutive pairs of straight-line elements in the same plane which is obtained for a cone.

Figure 3. shows the true development obtained for polyhedrons and single curved surface

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. Wrapped surfaces do not produce true developments, because pairs of consecutive straight-line elements do not form a plane. Also double-curved surfaces, such as a sphere do not produce true developments, because they do not contain any straight lines. An example of the approximate development of a sphere is shown in figure 4 .

Figure 4 showing an approximate development of a sphere.

Classifications of developments
1. Parallel-line development: They are made from common solids that are composed of parallel lateral edges or elements. e.g. Prisms and cylinders as shown in figure 5. The cylinder is positioned such that one element lies on the development plane. The cylinder is then unrolled until it is flat on the development plane. The base and top of the cylinder are circles, with a circumference equal to the length of the development. All elements of the cylinder are parallel and are perpendicular to the base and the top. When cylinders are developed, all elements are parallel and any perpendicular section appears as a stretch-out line that is perpendicular to the elements.

Figure 5 shows the parallel line development technique for (a) cylinder and (b) rectangular block.

2. Radial-line development
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. Figure 6 shows the radial development for a cone. The cone is positioned such that one element lies on the development plane. The cone is then unrolled until it is flat on the development plane. One end of all the elements is at the vertex of the cone. The other ends describe a curved line. The base of the cone is a circle, with a circumference equal to the length of the curved line.

Figure 6 shows the radial development method for a cone.

3. Triangulation developments:
Made from polyhedrons, single-curved surfaces, and wrapped surfaces. The development involve subdividing any ruled surface into a series of triangular areas. If each side of every triangle is true length, any number of triangles can be connected into a flat plane to form a development. This is illustrated in figure 7 for a triangular pyramid. Triangulation for single curved surfaces increases in accuracy through the use of smaller and more numerous triangles. Triangulation developments of wrapped surfaces produces only approximate of those surfaces.

Figure 7 shows the triangulation method for obtaining the development of a triangular pyramid.

4.Approximate developments
Approximate developments are used for double curved surfaces, such as spheres. Approximate developments are constructed through the use of conical sections of the object. The material of the object is then stretched through various machine applications to produce the development of the object. This is illustrated in figure 4.

Parallel-line developments
Developments of objects with parallel elements or parallel lateral edges begins 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. Figure 8 illustrates the steps followed for obtaining the development of a rectangular prism by parallel line development. In the front view, all lateral edges of the prism appear parallel to each other and are true length. The lateral edges are also true length in the development. The length, or the stretch-out, of the development is equal to the true distance around a right section of the object.

Figure 8. Stepwise procedure for obtaining the development of a rectangular prism.

Step 1. To start the development, draw the stretch-out line in the front view, along the base of the prism and equal in length to the perimeter of the prism. Draw another line in the front view along the top of the prism and equal in length to the stretch-out line. Draw vertical lines between the ends of the two lines, to create the rectangular pattern of the prism.
Step 2. Locate the fold line on the pattern by transferring distances along the stretch-out line in length to the sides of the prism, 1-2, 2-3, 3-4, 4-1. Draw thin, dashed vertical lines from points 2, 3, and 4 to represent the fold lines. Add the bottom and top surfaces of the prism to the development, taking measurements from the top view. Add the seam to one end of the development and the bottom and top.

Development of a truncated prism

The methoid of obtaining the development of a truncated prism is shown in figure 1.

Figure1. shows the development of a truncated rectangular prism.

Step 1: Draw the stretch-out line in the front view, along the base of the prism and equal in length to the perimeter of the prism.
Locate the fold lines on the pattern along the stretch-out line equal in length to the sides of the prism, 1-2, 2-3, 3-4, and 4-1.
Draw perpendicular construction lines at each of these points.
Project the points 1, 2, 3, and 4 from the front view
Step 2: Darken lines 1-2-3 and 4-1. Construct the bottom and top, as shown and add the seam to one end of the development and the top and bottom

Development of a right circular cylinder

The Procedure for obtaining the development of a cylinder is illustrated in figure 2.

Figure 2. Shows the step wise procedure for obtaining the development of a cylinder.

Step 1. In the front view, draw the stretch-out line aligned with the base of the cylinder and equal in length to the circumference of the base circle.
At each end of this line, construct vertical lines equal in length to the height of the cylinder.
Step 2. Add the seam to the right end of the development, and add the bottom and top circles.

3.Development of a truncated right circular cylinder

The development of a truncated cylinder is shown in figure 3.

Step1. The top view and front view of the cylinder is drawn. The stretch out line, is aligned with the base in the F.V., is drawn with length equal to the circumference of the cylinder. Construct the rectangle with the stretch out line as one length and height of the cylinder as the width.

Step2. The top circular view of the cylinder then divided into a number of equal parts . The stretch-out line is also divided into 12 equal parts from which vertical lines 1, 2, 3, 4, …. 12 are constructed. The intersection points in the T.V. are projected into the F.V. . Draw horizontal projectors from points 1, 2, 3, …., 12 to intersect the vertical lines 1,2, 3, 4, …. 12 in the stretch out line. , where the projected lines intersect the angled edge view of the truncated surface of the cylinder. The intersections between these projections and the vertical lines constructed from the stretch-out line are points along the curve representing the top line of the truncated cylinder. Join the intersection points with a smooth curve to obtain the developmentof the lateral surface of the cylinder.

Figure 3. shows the development of a truncated cylinder.

Step 3. Draw the circle with diameter equal to the diameter of the cylinder at any point on the base of the development to obtain the development of the base surface of the cylinder. Draw an ellipse with the truncated length (length 1-7 in the step 2) as major diameter and diameter of the cylinder as the minor diameter on the top part of the development to obtain the final development of the surfaces of the truncated cylinder.

4.Development of a right circular cone

The development of a cone is shown in figure 4. For a cone, the front view will be a triangle with the slant edge showing the true length of the generator of the cone. To begin this development, use a true-length element of the cone as the radius for an arc and as one side of the development. Draw an arc whose length is equal to the circumference of the base of the cone. This can also be determined by angle θ = (r/l) * 360°, where, r is the radius of the base of the cone and l is the true length of the slant edge. Draw another line from the end of the arc to the apex and draw the circular base to complete the development.

Figure 4. Development of a right cone.

5.Development of a truncated cone:

A cone of base diameter 40 mm and slant height 60 mm is kept on the ground on its base. An AIP inclined at 45° to the HP cuts the cone through the midpoint of the axis. Draw the development.

Solution:
The development of the truncated cone is shown in figure 5.

Figure 5. Development of the truncated cone (problem 5).

Draw the Front view and top view of the cone. Dive the circumference of the circle (Top View) in to 12 equal parts 1, 2, 3, 4, …., 12. Project these points on the Front view to obtain the points 1’, 2’, 3’, …., 12’. Draw a line inclined at 45 ° to the horizontal and passing through the mid pint of the axis of the cone to represent the AIP. The locate the intersection points of the AIP with the generators O’-1’, O’-2’, …. O’-12’ as P1’, p2’, p3’, …. P12’. Draw the projection (figure shown on the right of the Front view) by drawing the line O1 parallel to O’ 7’. Obtain the included angle of the sector. θ = (20/60)* 360 = 120° (following the procedure shown in problem 4). Then draw sector O–1–1– O with O as a centre and included angle 120°. Divide the sector into 12 equal parts (i.e., 10° each). Draw lines O–2, O–3, O–4, …, O–12. Draw horizontal projectors from P1’, P2’, …., P12’ such that it meets the line O1 at p1, p2, p3, …, p12. With O as centre and radius O’P1’, mark point P1 on line O1. With O as centre and radius equal to Op2, draw an arc to intersect the radial line O2 at point P2. Similarly obtain points P3, P4, …, P12, and P1. Join points P1, P2, P3, …., P12 and P1 to obtain the development of the truncated cone.

Development of Transition pieces
In industries when different pipes or vessels having different shapes and sizes are to be joined. These are facilitated by using special sections called transition pieces. Transition pieces are the sheet metal objects used for connecting pipes or openings either of different shapes of cross sections or of same cross sections but not arranged in identical positions. In majority of the cases, the transition pieces are composed of plane surfaces and conical shapes. The conical surfaces are developed by triangulation technique. These are highlighted in the subsequent paragraphs.

Triangulation development
The triangulation development is employed to obtain the development of transition pieces. These consists of the following:

Transition pieces joining a curved cross section to a non curved cross section (e,g, Square to round, hexagon to round , square to ellipse, etc.)
Joining two non-curved cross sections (e.g. square to hexagon, square to rectangle, square to square in un-identical positions)
Joining only two curve sections (e.g. Circle to oval, circle to an ellipse, etc)

In this method, the lateral surfaces of the transition pieces are divided in to a number of triangles. By finding the true lengths of the sides of each triangle, the development is drawn by laying each one of the triangles in their true shapes adjoining each other.

Transition pieces joining curved to Non-curved cross sections
Figure 1 shows the top view and pictorial view of two transition pieces: (a) the pentagonal base joined to a circular top and (b) circular base connected to a square top. The lateral surface of the transition piece must be divided in to curved and non-curved triangles as shown in figure 1. . Divide the curved cross section in to a number of equal parts equal to the number of sides of non-curved cross-section. Division points on the curved cross section are obtained by drawing bisectors of each side of the non-curved cross section. The division points thus obtained when connected to the ends of the respective sides of the non-curved cross-section produces plane triangles. In between two plane triangles there lies a curved triangle. After dividing in to a number of triangles, the development is drawn by triangulation method.

Figure 1. Shows the top view and pictorial vies of two transition pieces.

Development of a transition piece from square to circular section

Figure 2 shows the development of a transition piece connecting a circular top section with a square bottom section. Divide the circumference of the circle in to four equal parts by drawing perpendicular bisectors of the base edge (square edge) in the top view. Join 1-a, a-5, 5-b, b-9, 9-c, c-13, 13-d, d-1. The transition piece now consists of 4 plane and 4 curved triangles.1da, 5ab, 9bc, and 13cd are plane triangles and 1a5, 5b9, 9c13 and 13d1 are curved triangles.
Since the transition piece is symmetrical about the horizontal axis pq in the top view, the development is drawn only for one half of the transition piece. The front semicircle in the top view is divided in to eight equal parts 1,2,3,4, etc. Connect points 1,2,3,4 and 5 to point a. Project points 1,2,3,etc to the front view to 1’,2’,3’. etc. Connect 1’, 2’, 3’ etc to a’ and 5’, 6’, 7’, 8’ 9’ to b’. The development of the one half of the transition piece is drawn from the true length diagram. The procedure for drawing the true length diagram is explained below.

Figure 2. Development of a transition piece with circular top and square bottom.

True length diagram and development

Draw vertical line XY.
The first triangle to be drawn is 1pa (shown in the top view)
The true length of sides 1p and 1a are found from the true length diagram. To obtain true length of sides 1p and 1a, step off the distances 1p and 1a on the horizontal drawn through X to get the point 1P’ and 1A’. Connect these two points to Y. The length Y-1P’ and Y-1A’ are the true lengths of the sides 1p and 1a respectively.
1₁P = Y-1P’. Draw a line with center 1₁ and radius Y-1A’. With P as center and radius pa, as measured from the top view, draw an arc to cut to meet at A.
With A as center and radius equal to true length of the line 2a (i.e Y-2A’), draw an arc.
With 1₁ as center and radius equal to 1-2 (T.V), draw another arc intersecting the pervious arc at 2₁.
Similarly determine the points 3₁, 4₁ and 5₁.
A -1₁-2₁-3₁- 4₁- 5₁ is the development of the curved triangle 1-a-5.
A-5₁-B is the true length of the plain triangle a-5-b.
Similar procedure is repeated for the other three curved triangles and plain triangles.

2.Development of a square to hexagon transition piece

Figure 3 shown the development of a square to hexagon transition piece. The front view , Top view and the true length diagram for half of the transition piece is shown.
The transition piece is assumed to cut along PQ. Triangles 1-p-a and 1-a-2 and trapezium a-2-3-b are obtained.
To develop the lateral surface a-2-3-b, it is divided in to two triangles by connecting either a-3 or 2-b and completed by triangulation method as detailed in the above problem.

Figure 3. Dvelopment of a square to hexagon transition piece

3.Development of transition pieces joining two curved surfaces

The development of a transition piece of circle to circle but with different diameters is shown in figure 4, where the axes of the two circular sections are offset by 15 mm. first draw the TV and FV of conical reducing piece. Divide the two circles in to twelve equal parts. Connect point 1-a,2-b, 3-c, etc in the TV and 1’-a’, 2’-b’,etc in the FV. These lines are called radial lines. The radial lines divide the lateral surface in to a number of equal quadrilaterals. Their diagonals are connected (dashed lines) forming a number of triangles. The true length diagram are drawn separately for radial and diagonal lines and is shown in figure 4. The development is drawn for half of the transition piece by triangulation method and using the true length of the radial lines and diagonal lines.

Figure 4. Development Transition pieces joining two circular sections.