IT is important to be able to find the true length of a line in some aspects of Engineering Drawing, particularly in the development, or laying out in a plane, of sheet metal details. It was stated in Chapter 5 on orthographic projection that the projection of a line will only show its true length if the line is parallel to the plane on which it is projected.
This is illustrated in Figures I (a), (b) aRd (c) opposite.
In Figure I (a) the plan a2 b2 of the line AB is parallel to the XY line.
Hence the line is parallel to the vertical plane of projection, and its elevation a1bIon this plane is a true length. Also the angle 8 is the true inclination of the line to the horizontal plane.
The line AB in Figure I (b) is parallel to the horizontal plane. In this case, therefore, the plan a2b2 of the line shows its true length, and the angle 1>its true inclination to the vertical plane.
Figure l(c) shows the elevation al bI of the line AB to be parallel to the auxiliary vertical plane, and so the projection of the line on this plane gives its true length. Angle 8 is the true inclination of the line to the horizontal plane.
When a line is not parallel to one of the normal planes of projection two methods may be used to find its true length. The line may be moved until it is parallel to a principal plane, or an auxiliary plane may be used which is parallel to the plan or elevation of the line. Figure 2 illustrates the first of these approaches, which is called the revolution method.
In both illustrations the end A of the line remains stationary and the end B is revolved until the line is parallel to either the horizontal or vertical plane. Thus the line sweeps out a right cone with its apex at A.
In one view B moves in a circle; in the other it moves along a straight line. There are two positions for the true length, depending on the direction of rotation of B. If the true length appears in the elevation, then the angle it makes with the XY line is the true inclination of the line to the horizontal plane. If it appears in the plan, its angle to XY is the true inclination to the vertical plane, since, relative to the plan, the XY line represents the vertical plane, and vice versa.
Finding the true length of a line by triangulation, as in Figure 3, is essentially the same as the revolution method, but for some purposes it is more convenient. The plan length a2 b2 of the line is set off at right
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angles to the difference in vertical height of the ends of the line in elevation. The hypotenuse AB of the right angled triangle so formed is the true length of the line, and angle 8 is the true inclination to the horizontal plane. The true length will also be found as the hypotenuse of a right angled triangle formed by setting off the elevation al bl at right angles to the difference in depth of the ends of the line in plan. In this triangle the true inclination of the line to the vertical plane will appear.
The second approach, using auxiliary planes, is shown in Figure 4.
The plan a2b2 of the line is viewed normally in the direction of arrow S, and a view in this direction is drawn on an auxiliary vertical plane Xl YI. This plane is parallel to a2b2, so the auxiliary elevation drawn on it will show the true length of AB. The heights h and hI are the same in both elevations, and 8 is the true inclination of the line to the horizontal plane.
If the elevation aI bl of the line is viewed normally in the direction of arrow T, an auxiliary plan projected on Xl Y\ which is parallel to al b\
will again show the true length of the line. The widths wand WI from the original plan are the same in the auxiliary plan. cp is the true inclina-tion of the line to the vertical plane. For a fuller treatment of the prin-ciples of projecting auxiliary views the student is referred to Book 2.
In the worked examples which follow the methods for finding the true length of a line have been used to draw orthographic views of lines and flat laminae in various positions relative to the planes of projection.
Example 1
Sufficient information is given to draw three orthographic views of the lines. The true lengths of AB and BC have been found by the revolution method and that of AC by triangulation. Since the three true lengths all appear in elevations, the angles they make with the XY line are their true inclinations to the horizontal plane.
Example 2
AB can be positioned in the three views directly from the given data.
Its front elevation al bl is true length. To find cl imagine the triangle to be rotated about al bl until it is parallel to the vertical plane. In this position it can be drawn, since it will show its true size and shape al bl c4. When the triangle is rotated about al b\ c4 will appear to move along a line perpendicular to al bl. Where this line meets the horizontal plane is the position of cl. Project from c1 into the plan and fix c2on the projector by striking an arc from a2 equal to a1c4, the true length of AC.
This can be done because AC is a true length in plan. The drawing can now be completed by positioning c3in the end view.
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Example 3
The plan view of the quadrant, which is a straight line, can be drawn from the given information. The elevation and end view project as quarter ellipses, and to draw them an auxiliary view in the direction of arrow X is required. This view shows the true shape of one quadrant.
On the true shape select points such as a4 and b4, project them to the plan at a2and b2, and then to the elevation at aI and b1• The heights h and hI from the auxiliary view fix al and bl on the projectors from the plan. Widths wand WIfrom the plan position a3and b3in the end view.
The elevation and end view can then be lined in with a french curve.
TR UE LENGTH PROBLEMS
Draw full size using First or Third Angle projection as required by the question.
1 Two lines AB and BC are shown with A on the vertical plane, B on the auxiliary vertical plane and C on the horizontal plane. Draw three views in First Angle projection of the lines on these planes, and determine the true length of AB, AC and BC and the true angle 8.
2 The triangle is positioned with AC on the horizontal plane and B against the vertical plane. Draw three views in First Angle projec-tion of the triangle on the given planes.
3 The given triangle has A on the vertical plane, B on the auxiliary vertical plane and C on the horizontal plane. Project views of the triangle on these planes, using First Angle projection.
4 The quadrant is positioned as shown with AB on the vertical plane, parallel to the horizontal plane. Draw views of the quadrant on the planes using First Angle projection.
5 The triangle has BC on the horizontal plane and A on the auxiliary vertical plane. Draw a front elevation, plan and end view in First Angle projection of the triangle in the given position.
6 Using First Angle projection draw three views of the semicircle.
A is on the auxiliary vertical plane, B is on the horizontal plane, and the semicircle touches the vertical plane.
7 The plan view of a piece of plate is given, the plate being bent along the line Be. BC lies on the horizontal plane and D is 20 mm above the horizontal plane. The true sizes of angles Band Care 60° and 30° respectively. Use Third Angle projection to draw the given plan and an elevation looking in the direction of arrow R. Determine the
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shape of the plate before bending and the true inclination of the edge AB to the horizontal plane.
8 A piece of sheet metal is bent as shown and has points D and B on the horizontal plane, A on the vertical plane and C on the auxiliary vertical plane. Draw three views of the plate on the given planes in First Angle projection, and construct its shape before bending.
9 Three orthographic views of a line are given in the figure. Draw these views and position a point C in each of them, such that the true lengths of AC and BC are 82 mm and 95 mm respectively, with C on the horizontal plane. Use Third Angle projection.
10 The plan and elevation of a triangular pyramid are given. Draw these views in Third Angle projection and find the true lengths of AB, AC and AD, the true inclinations of AB and AC to the hori-zontal plane, and the true inclination of AD to the vertical plane.
11 Using Third Angle projection reproduce the three given views of the triangle.
]2 Views are given of a sheet metal component with an open top and bottom. Draw these views in Third Angle projection and construct the shape of half the component before it is bent.