## Developments and Intersections

This is an aspect of workshop drawing that often arises when, for example, we need to cut out sheet metal which can be rolled into a specific shape, or when we need to cut a hole in a container - boiler or tank -which will be a close fit to a branching piece. A development is no more than the shape of the piece of sheet in the flat before folding and jointing. An intersection is the profile of the curve formed when two sections meet. You will find six of these on every nut where the cone of the chamfer meets the six flats. Normally we do not need to draw these accurately, but there can be occasions when this is necessary.

It is more than unfortunate that development drawings are usually a maze of lines. They can and do frighten off those not accustomed to them, and give the impression that this is very difficult; quite the wrong impression. Gypsy tinsmiths 100 years ago could lay out a development of (e.g.) a kettle spout almost in their sleep. And, of course, though there may be a maze of lines they are all drawn one at a time, and most will be a simple repetition of that drawn previously, but in a new place. However, there is an easy way to deal with this problem. In each of the examples I am going to show toy how to do the job yourself as you read the page. Make a neat sketch of the object, and follow the instructions. Then, if you have time, get out the drawing-board and have a go on a similar problem of your own.

There can be scores of problems both for development and for intersection curves - indeed, there is at least one book on the subject dealing with nothing else. But the principle involved is the same in each case, and once you have mastered this you will be able to progress under your own steam to more elaborate systems should any come your way. But there is one matter which is quite vital, and which I must deal with first and that is the concept of the true length. Look at Fig. 76. Here we have a cone with a point 'X' on its side (it could be the corner of a hole). We want to mark this point on the flat sheet before we form the cone, so we need to know the length OX. However, looking at the elevation (the side view) we cannot scale the drawing because the line OX is sloping away from the eye; similarly, we cannot use the plan view, for the same reason. We can, of course, work

it out very quickly, for the truly vertical distance, OX is one side of a right-angled triangle and the radial line in plan O'X', is its base. So we can draw the triangle I have shown as abc and either measure ac or work it out from ac2= ab2 + be2. However, there is a simpler way. Look at the perspective sketch of the cone I have drawn. You will see that if we rotate the position of X about the vertical axis of the cone to the position X" then OX" gives the length we require. Looking back at the main drawing you will see that I have done just that; follow the arc from X' in the plan, then project upwards until we meet the side of the cone at X". Simple, isn't it? Simpler still when you notice that a line projected through X at right angles to the axis of the cone also arrives at the right place. In most of what follows we shall be using this procedure, or some variant of it, no matter how complex it may seem. I am going to start with the cone, as this crops up very often indeed - for funnels in the shop and for lampshades in the house, too.

Development of a cone (Fig. 77)

First a few definitions. The vertical height is that measured along the axis or centreline. The slant height is that measured along the sloping side. The cone angle is the angle included at the peak of the cone, and the developed angle is that made at the peak of the sheet in the flat. The base diameter is the diameter of the complete cone; we shall, shortly, be dealing

Fig. 77 The development of a cone onto a flat sheet.

with the case where the cone is cut off at an angle.

The development is a very simple one, forming, as you see, a fan shape. In the drawing I have supposed that we are making a funnel, so that we need a hole in the top of diameter'd'. It needs little abstruse mathematics to see that the length of the side of the fan must be S-s' - the difference between the slant height of the main cone and the little cone we do not want. It is also clear that the length of the arc at the extremity of the fan must be equal to the circumference of the cone base diameter, which will be nD (n= 22/7 or 3.14 for this sort of work). However, this is not easy to lay out, and it is easier to work out the developed angle. I have given the formula on the drawing. The upper one is the easier to use, and if you don't know what the slant height'S' is you can either work it out from S2= H2 + (1/2 D )2 (but note the 'S' must be taken from the vertex 'O') or you can measure it. I always draw these developments to as large a scale as I can and measure from the drawing.

You will, of course, require a small overlap to make the joint -1 have shown a suggestion on the drawing. Although this is not a treatise on how to make cones, if you do need one with a true point then I suggest the method shown in Fig. 78. It is