Isometric Drawing Of A Rectangular Prism

17 Isometric Projections

After reading this chapter, you will be able to

Draw a pictorial or an isometric projection from orthographic projections of simple solids or objects bounded by plane surfaces
Draw a pictorial or an isometric projection from orthographic projections of simple solids or machine parts that are created by the addition and subtraction of other simple solids

17.1 INTRODUCTION

Multiview orthographic projections generally show the lengths of an object along only two of its principal axes in any particular view. The length along the third axis is not visible in the same view. This makes it difficult to interpret the views, and only technically trained persons can interpret the meaning of these views.

On the other hand, pictorial projections can easily be understood even by persons without any technical training because such views show all the three principal axes of an object in the same view. A pictorial view may not show the true shape and size of any principal surface of an object, but as all the principal faces are seen in the same view, it is convenient for even untrained persons to imagine the shape of the object when given a pictorial view. Hence, pictorial views, and especially isometric projections, which are a type of pictorial views, are extensively used in sales literature.

In this chapter, the discussion is limited to the drawing of isometric projections of uncut simple solids and simple machine parts using the box method.

17.2 TYPES OF PICTORIAL PROJECTIONS

There are various types of pictorial projections but only the following, which are extensively used by engineers, are discussed in this book:

Isometric projections
Oblique parallel projections
Perspective projections

Isometric projections are discussed in this chapter.

17.3 ISOMETRIC PROJECTIONS

An isometric projection is an orthographic projection that is obtained in such a way that all the principal axes are projected in the same view and the reduction in their lengths is in the same proportion. For this purpose, the object is placed in such a way that its principal axes are equally inclined to the plane of projection.

The projections of a cube are shown with the solid diagonal of the cube parallel to both the HP and the VP in Figure 17.1 (a). In this position, the solid diagonal A ₁ C is perpendicular to the profile plane (PP) and the principal edges of the cube are equally inclined to the PP. Hence, in the side view, the lengths of all the principal edges are equally shortened. The side view in this position represents the isometric projection of the cube.

Figure 17.1 (b) represents the usual orthographic projections of a cube with each of the principal edges perpendicular to one of the principal planes of projections in the following ways:

AB, CD, A ₁ B ₁ and C ₁ D ₁ are perpendicular to the VP, and hence are vertical lines in the top view.
AA ₁, BB ₁, CC ₁ and DD ₁ are perpendicular to the HP, and hence are vertical lines in the front view.
BC, B ₁ C ₁, AD and A ₁ D ₁ are perpendicular to the PP, and hence are horizontal lines in the front as well as the top views.

In isometric projections in the side view, a″b″, c″d″, a″₁ b″₁, and c″₁ d″₁ are all parallel to each other and inclined at 30° to the horizontal in the same direction.

FIGURE 17.1 The isometric projection of a cube

Similarly, b″c″, d″a″, b″₁ c″₁, and d″₁ a″₁ are all parallel to each other and inclined at 30o to the horizontal in the other direction. The lines a″b″₁, b″b″₁, c″c″₁ and d″d″₁ are all parallel to each other and are vertical lines. The lengths of all these lines, which are equal on the object, are equal in the side view also but they are isometric lengths and not the original true lengths. The foregoing discussion leads to the conclusions tabulated in Table 17.1.

If, while drawing isometric projection, the true lengths are used instead of isometric lengths for principal lines, the drawing's shape will remain the same, but it will be a little oversize. Such drawings are known as isometric drawings instead of isometric projections. If a cube is added on one of the faces of another cube drawn in an isometric view, the shape will be that of a square prism, as shown in Figure 17.2 (a). Similarly, if more cubes are added in proper positions, the shape will be that of a rectangular prism, as shown in Figure 17.2 (b). Thus, a rectangular prism can be drawn by drawing principal lines in positions given in Table 17.1 [see Figure 17.2 (c)]. It may be noted that only the principal lines have fixed positions and particular isometric lengths. The other lines are drawn by locating their endpoints with the help of coordinates measured in the principal directions. In an isometric projection, the three principal edges of a rectangular object are inclined at 1200 to each other but their orientation can be as shown in Figure 17.2 (d) or 17.3.

TABLE 17.1 Isometric and orthographic projections of principal lines

Depending upon how the cube is tilted, these positions can change, but the principal edges, which are mutually perpendicular on the object, will always remain inclined at 120° to each other in the isometric projection [see Figures 17.3 (a)–(d)]. However, in this textbook, the positions indicated in Table 17.1, are generally adopted.

17.3.1 The Isometric Scale

As can be observed from Figure 17.1 (a), the edge ab in Step I will be representing the true length but in the side view (in Step II), which is an isometric projection, the length of a″b″ is not the true length. Looking at the positions of these two lines, the relation between the true length of the principal line on the object and its length in the isometric projection can be established. The length in isometric projection is always times the true length for all principal edges. These lengths can be obtained by preparing an isometric scale as follows:

FIGURE 17.2 The isometric projections of prisms

As shown in Figure 17.4, draw (i) a horizontal line; (ii) a line inclined at 30° to the horizontal, and (iii) a line inclined at 45° to the horizontal. Plot the measurements from an ordinary scale along the 45° line and from each division point, draw vertical lines to intersect the 30° line. Divisions obtained on the 30° line gives the isometric lengths to the same scale as the ordinary scale used for plotting the measurements on the 45° line. This means that if measurements from an ordinary 1:2 scale are plotted on the 45° line, the corresponding measurements on the 30° line will give the isometric lengths on a 1:2 scale.

It is advisable that 1 cm lengths be plotted for the full length of the scale and 1 mm divisions be plotted at the common starting point of all the three lines on the extreme left side so that any measurement can be taken with an accuracy of 1 mm. For better accuracy, a small portion near the common point should be left unused for marking divisions because the 30° and 45° lines have a very small gap between them at that end.

FIGURE 17.3 Isometric projections with different visible faces

17.3.2 Drawing Isometric Projections of Solid Objects

Isometric projections can be drawn, as explained earlier, only of objects with mutually perpendicular principal edges. For such objects, principal edges can be drawn in fixed positions and with the lengths reduced to times the original. For a solid object that does not have such mutually perpendicular edges the box method is generally used to draw the projections. The object is imagined to have been placed in a transparent rectangular box that it fits into exactly. The isometric projection of such a box can always be drawn, and then the required points on the boundaries of surfaces of the object are located with the help of the coordinates measured in the principal directions. This is shown in Figure 17.5 where a pentagonal prism is imagined to be placed in a transparent rectangular box and (a) shows the orthographic projections while (b) shows the isometric projections.

FIGURE 17.4 An isometric scale

FIGURE 17.5 (a) The orthographic projections of a pentagonal prism (b) The isometric projections of a pentagonal prism

In Figure 17.5 (a) the projections are enclosed within exactly fitting rectangles, which are the principal views of a rectangular box positioned with each of its sides either vertical or horizontal. P, Q, R and S are the corners of the rectangular box at the bottom and P ₁, Q ₁, R ₁ and S ₁ are the corners at the top, and their front and top views are named accordingly.

The box edges PP ₁, QQ ₁, RR ₁ and SS ₁ are perpendicular to the HP and are projected as vertical lines in the FV as well as in the isometric projection.

QR, Q ₁ R ₁, PS and P ₁ S ₁ are perpendicular to the VP and are projected as vertical lines in the TV and lines inclined at 30° to the horizontal in the isometric projection.

PQ, P ₁ Q ₁, RS and R ₁ S ₁ are perpendicular to the PP and are projected as horizontal lines in the FV and the TV and as lines inclined at 30° to the horizontal in the other direction compared to those perpendicular to the VP.

The lowest point O in the isometric view where the three mutually perpendicular edges of the object, PQ, QR and QQ ₁, meet is generally known as the origin point.

To locate any point of the object in the isometric view, start from any known point of the enclosing box (preferably from the one nearer to the required point) and measure coordinates in the horizontal and vertical directions in the FV and the TV. Note that the measurements in the horizontal and vertical directions in the FV are perpendicular to the PP and the HP, respectively. Similarly, the horizontal and vertical measurements in the top view are perpendicular to the PP and the VP, respectively. If the orthographic projections are drawn to an ordinary scale, the coordinate distances measured from them are reduced to the isometric scale and then plotted in the isometric projection in the proper direction.

In Figure 17.5 (b), to locate M ₁ in the isometric view, the coordinates are measured starting from q ₁ in the top view and q′₁ in the front view. The distance x in the FV and the TV is same in the direction perpendicular to the PP and should be plotted only once in the isometric view. The distance y is perpendicular to the VP. These coordinates are shown plotted accordingly in the isometric projection (see Figure 17.5 (a). The corner points of the prism are plotted in the same way.

To determine visibility, draw surfaces sequentially starting from those touching or nearest to the visible faces of the enclosing box. Thus, the surface A ₁ B ₁ C ₁ D ₁ E ₁ that is touching the top face is fully visible. The surfaces DD ₁ E ₁ E, CC ₁ D ₁ D and AA ₁ E ₁ E are nearest to visible front and side faces of enclosing box, and hence they are visible. The surfaces ABCDE, BB ₁ C ₁ C, and AA ₁ B ₁ B come later when looking from the direction of observation and are hidden as they are covered by the previously drawn surfaces. Hence, the lines AB, BC and BB ₁ are drawn by short dashed lines.

17.3.3 Procedure for Drawing Isometric Projections of an Object

The steps for drawing isometric projections of an object are as follows:

Step I:	Draw the orthographic projections of the given object and enclose each view in the smallest possible rectangle, as shown in Figure 17.5 (a). The sides of the rectangles should be either vertical or horizontal lines because they are supposed to be the principal lines of the enclosing box of the object.
Step II:	Select the faces that are to be visible in such a way that the maximum number of visible lines/surfaces is obtained in the isometric projection. Generally, the front face, the top face, and one side face are made visible. If the left-side view gives the maximum number of visible lines, the left face is made visible. If the right-side view gives the maximum number of visible lines, the right face is made visible. Accordingly, the enclosing box is drawn by thin lines, the position of the lines being the same as those given in Table 17.1 and lengths reduced to the isometric scale.
Step III:	Correlate the projections of the various surfaces in all the views by using the properties of projections of the same plane surfaces given in Chapter 16. Having correlated the projections in two views or more, points should be measured in principal directions in any two views and should be plotted in isometric projections in the directions given in Table 17.1. Coordinate distances should be reduced to the isometric scale before plotting.
Step IV:	Draw the boundaries of all the surfaces by appropriate conventional lines depending upon their visibility.

Now, let us look at some examples.

Example 17.1 Draw the isometric drawing of a cylinder of 40 mm diameter and 55 mm length resting on its base.

Solution (Figure 17.6):

FIGURE 17.6 Solution of Example 17.1

Draw the orthographic projections of the cylinder, and then draw by thin lines the enclosing rectangles of the two views, as shown in Figure 17.6 (a).
Draw the enclosing rectangular box by thin lines. All the measurements in the principal directions are to be reduced to the isometric scale, if the isometric projection is to be drawn. If true lengths are used, the drawing will be slightly oversize and is known as an isometric drawing.
A number of points are selected on the circular edges, which are boundaries of surfaces. The related projections in the FV are indicated by the same lowercase letters, each followed by the prime symbol.
The points can be plotted by measuring the necessary coordinates, as shown. (Each coordinate length is reduced to the isometric scale before plotting if the isometric projection is to be drawn.)
Projections can now be completed by drawing appropriate conventional lines for all the boundaries of surfaces. The generators DD ₁ and HH ₁ are on the boundary of the cylindrical surface. They are projected and, therefore, drawn by thick lines as they are visible [see Figure 17.6 (b)].

In isometric projections, plotting points on the circular edge of an object by the coordinate method is a time-consuming process. As circles are frequently required to be projected in isometric projections, an approximate method known as the four-centre method is generally used to get an approximate elliptical shape, as shown in Figure 17.6 (c).

The enclosing square of the circle in the orthographic view is drawn as a rhombus in the isometric view. Now, perpendicular bisectors of all the four sides are drawn and the points of intersection of these bisectors are the required four centers, with the help of which four arcs can be drawn. These give an approximate shape of an ellipse touching the four sides of the rhombus at midpoints [see Figure 17.6 (c)].

Example 17.2 Draw the isometric drawing of a cone of diameter 40 mm and length of the axis 55 mm, when it is resting on its base.

Solution (Figure 17.7):

FIGURE 17.7 Solution of Example 17.2

Draw the orthographic projections and enclose each view in a rectangle with vertical and horizontal lines, as shown in Figure 17.7 (a).
Draw the enclosing rectangular box by thin lines. Reduce the lengths of principal lines to isometric scale if isometric projection is to be drawn.
Draw by thin lines the base circle as an ellipse using the four-centre method. Locate the apex with the help of two coordinates x and y from the corner P ₁.
Complete the projections by drawing appropriate conventional lines for the base circle and the two generators tangent to the ellipse to represent the boundary of conical surface, as shown in Figure 17.7 (b). Note that a part of the base circle will not be visible.

Example 17.3 Draw the isometric drawing of the semicircular-cum-rectangular plate shown in two views in Figure 17.8 (a).

Solution [Figure 17.8 (b)]:

Half of the object is a cylinder, and the other half is a prism. Only two centres (of the four-centre method explained in Example 17.1) are required to be located for drawing each semicircle as a semi-ellipse in the isometric drawing. The figure is self-explanatory. Note that, normally, hidden lines are not drawn in isometric drawings.

Example 17.4 Draw the isometric drawing of the isosceles triangular plate rounded at the vertex, as shown in the two views in Figure 17.9 (a).

Solution [Figure 17.9 (b)]:

Draw the circular portion at the vertex, initially, as a semi-ellipse in the isometric drawing.
Locate the base edge lines and then draw tangents to the semi-ellipse through points A and B.
Remove the extra length of the curve, if any, beyond the points of tangency and complete the isometric drawing.

Example 17.5 Draw the isometric drawing of the triangular plate rounded at both ends, as shown in the two views in Figure 17.10 (a).

FIGURE 17.8 (a) Example 17.3 (b) Solution of Example 17.3

FIGURE 17.9 (a) Example 17.4 (b) Solution of Example 17.4

Solution [Figure 17.10 (b)]:

Initially, draw ellipses for the upper and lower circular parts.
Then, draw two tangents touching the two ellipses.
Draw the back face parallel to the front one.
Complete the isometric drawing as shown in Figure 17.10 (b).

FIGURE 17.10 (a) Example 17.5 (b) Solution of Example 17.5

Example 17.6 Draw the isometric drawing of the rectangular plate rounded at the top corners, as shown in the two views in Figure 17.11 (a).

Solution [Figure 17.11 (b)]:

When a quarter circle is to be drawn, it is not necessary to draw a complete rhombus enclosing the full ellipse in the isometric drawing.

From the corner points, locate the points of tangency A, B, C and D, each at radius distance and through these points, draw lines perpendicular to the respective sides.
The normal lines meet at points that are the required centres for drawing arcs in the isometric drawing. Draw the required arcs and complete the drawing.

Example 17.7 Draw the isometric drawing of the object shown in the two views in Figure 17.12 (a). Use the point O as the origin.

Analysis:

The object can be considered to be a rectangular block with a trapezoidal slot and a semicircular-cum-rectangular plate with a hole attached to it on one side. It may be noted that the trapezoidal slot has a total of ten edges, namely ab, bc, cd, a ₁ b ₁, b ₁ c ₁, c ₁ d ₁ in the top view and a′a ₁′, b′b ₁′, c′c ₁′ and d′d ₁′ in the front view for edges AB, BC, CD, A ₁ B ₁, B ₁ C ₁, C ₁ D ₁, AA ₁, BB ₁, CC ₁, and DD ₁.

FIGURE 17.11(a) Example 17.6 (b) Solution of Example 17.6

FIGURE 17.12 (a) Example 17.7 (b) Solution of Example 17.7

Solution [Figure 17.12 (b)]:

Draw the rectangular block.
Draw the trapezoidal slot.
Add the semicircular-cum-rectangular plate on the side of the block.
Add a circular hole in the plate.

It may be noted that lower circular edge of the hole will be seen partly through the hollow of the hole. Complete the projections, as shown in the Figure 17.12 (b).

FIGURE 17.13 (a) Example 17.8 (b) Solution of Example 17.8

Example 17.8 Draw the isometric drawing of the object shown in two views in Figure 17.13 (a).

Analysis:

The object is a semicircular-cum-rectangular plate at the bottom with a rectangular block having tapered surfaces on the two sides placed on it. Further, there is a semicircular-cum-rectangular cut in the plate and a channel formed in the block.

Solution [Figure 17.13 (b)]:

Draw the rectangular block in the isometric projection.
Add the semicircular-cum-rectangular plate.
Draw the channel in the block and the semicircular-cum-rectangular cut in the plate.

Note that the points D, E, F and G are located by coordinates. Further, observe that the lower edge of the semicircular-cum-rectangular cut is partly visible through the cut, as shown in Figure 17.13 (b).

Complete the isometric drawing as shown in the figure.

Example 17.9 Draw the isometric drawing of the object shown in Figure 17.14 (a).

Solution [Figure 17.14 (b)]:

Note that points on the curved line on the inclined surface are normally required to be plotted by measuring coordinates from orthographic projections. But, it is not necessary when solving such problems to measure the coordinate distances from orthographic projections. The required coordinate distances in X, Y and Z directions from a known corner or an edge of the enclosing box for points A, B and so on can be obtained from isometric drawing also, as shown in Figure 17.14 (b).

17.3.4 The Isometric Projection of a Sphere

As discussed earlier, an isometric projection is an orthographic projection obtained by placing the object with its mutually perpendicular principal axes equally inclined to the plane of projection. The orthographic projection of a sphere, placed in any position, is a circle with the radius equal to the true radius. If a plane parallel to the plane of projection cuts the sphere passing through the centre, the section will be a circle of the true radius. This circle appears as the boundary of a spherical surface to the observer and, being parallel to the plane of projection, is projected as a circle of true radius.

FIGURE 17.14 (a) Example 17.9 (b) Solution of Example 17.9

FIGURE 17.15 (a) Orthographic projections of a sphere (b) Isometric projections of a sphere

In Figure 17.15 (a), a sphere is shown resting on the frustum of a pyramid in the front and top views. The sphere is in contact with the top surface of the pyramid at the point c′. The distance between the centre of the sphere s′ and the point c′ is h′ which is equal to the true radius. The line s′c′ being vertical, that is, perpendicular to the HP, its isometric projection will be a vertical line with the distance SC equal to h′, reduced to the isometric scale. When a circle of the true radius is drawn to represent the sphere in the isometric view as shown in Figure 17.15 (b), the point of contact C remains within the circle. Thus, if the isometric drawing is drawn using the true lengths for all the principal lines, the sphere should be drawn as a circle with its radius increased in inverse proportion of the isometric scale. This is done so that the point of contact remains within the circle.

Example 17.10 Draw the isometric projection of the machine part shown in the two views in Figure 17.16 (a).

Solution [Figure 17.16 (b)]:

In the isometric projection, the spherical head should be drawn as a circle with the true radius while all other principal dimensions should be reduced to the isometric scale. The sphere where it meets the cylindrical surface of the handle will be a circle of diameter equal to the cylinder diameter. As it will not be visible, it is shown as an ellipse drawn by hidden lines. Normally, hidden lines are not drawn in isometric projections or drawings.

FIGURE 17.16 (a) Example 17.10

FIGURE 17.16 (b) Solution of Example 17.10

Draw the isometric drawing of the object shown in the two views in Figure E.17.1.

E.17.1
Three views of an object are shown in Figure E.17.2. Draw the isometric drawing of the object.

E.17.2
Three views of an object are given in Figure E.17.3. Correlate the projections of the various surfaces and draw the isometric drawing of the object.

E.17.3
Draw the isometric drawing of the object shown in Figure E.17.4.

E.17.4
Figure E.17.5 shows two views of an object. Draw the isometric drawing of the object.

E.17.5
Draw the isometric drawing of the object shown in the two views in Figure E.17.6.

E.17.6
Draw the isometric view of the object shown in Figure E.17.7.

E.17.7
Two views of an object are shown in Figure E.17.8. Correlate the projections of surfaces and draw the isometric view of the object.

E.17.8
Three views of a block-type object are shown in Figure E.17.9. Draw the isometric projections of the block.

E.17.9
The front, top and left-hand-side views of a block-type object are shown in Figure E.17.10. Draw the isometric drawing of the given block.

E.17.10

The front view and four different side views are given in Figure E.17.11. Draw isometric projections of the four given objects if the front view is common for all the four objects.

E.17.11
Two views of a socket are shown in Figure E.17.12. Draw the isometric drawing of the given socket.

E.17.12
Figure E.17.13 shows an object in two views. Draw the isometric view of the object.

E.17.13
The front and top views of an object are shown in Figure E.17.14. Draw the isometric view of the object.

E.17.14
Figure E.17.15 shows the front and left-hand-side views of an object. Draw the isometric view with O as the origin.

E.17.15
Figure E.17.16 shows the front and left-hand-side views of an object. Take O as the origin and draw the isometric drawing of the object.

E.17.16