Following are methods of copying a drawing to a scale - th of the original scale by means of an easily-constructed geometrical diagram, which may be found useful to mechanical and architectural draughtsmen - at any rate, in the absence of proportional compasses. Indeed, when m is less than 3, working with a diagram is at least as easy as working with the clumsy instrument just named, and the results are as accurate, supposing the diagram to be carefully constructed.
(a) In Fig. 1, let A and B be fixed points in a piece of cardboard or thick drawing paper, and AP, BP, any two lines drawn from them and meeting in P. Then, if in all cases A P = m . BP, the locus of P is a circle whose centre (T) is on A B produced.
To prove this, draw PN perpendicular to A B, or A B produced. Let B N = x N P = y, and consider the length of AB (the base) as representing unity. Then y2+(x + l)2 = m2 (y2 + x2) from which we derive - y2+(x- l/m2 - 1)2 = m2 (m2 + 12).
This is the equation to a circle. BT = 1/m2-1, and the radius TP = m/m2 - 1 By giving to m different values, we can calculate the corresponding values of B T and T P. Thus, if m = √3/√2, which is the ratio used in isometric projection, it will be found that BT = 2, and TB = 2.45. If m = 3/2, BT = 0.8, and TP = l.2. If m = 2, BT = 0.33, and TP = 0.67, etc. The difference between the shortest length, A D, that can be taken on the original drawing and A H, the longest, decreases as we increase m, and it is easily shown that the range of the instrument, in its most simple form, is expressed by the formula AH= m+1/m-1. AD. In practice it is rarely necessary to calculate numerically the lengths BT and TP, for by a simple geometrical construction we are enabled to produce a diagram to any ratio m, given two lines whose lengths are in that ratio. Thus, suppose AB to be a suitable length for the base, and A P to equal m. B P. Bisect the angle A P B by the line P D. Then, since, by Euclid VI. 3, AD:DB:: AP: BP .. A D: D B:: m: 1, which shows that P D is a chord of the required circle. Bisect PD in L, and draw LT perpendicular to P D. Evidently T, since it lies on both L T and A H, must be the centre, T P being the radius.
Describe the circle, and the diagram is complete.
The mode of using the diagram requires little explanation. A length A P, from the original drawing, being set off with ordinary compasses from A, determines the position of the point P, and we take P B for the corresponding length in our copy. To prevent their being holed by compass points, A and B ought to be protected by horn centres.
For any measurement shorter than A D, when it cannot be taken indirectly (that is, as the difference between two longer measurements), a supplementary circle, constructed to a shorter base than AB, is required. In the diagram, Fig. 2, from A draw any line, A A = A D, the shortest of the measurements; join H A, and through T, B, and D draw lines parallel to HA, meeting A h in t, b, and d respectively. Then, A 6 is the new base, and t the centre of its circle, of which the radius is t d. Supposing A B to be of sufficient length, a third circle and base can be derived from the second, a fourth from the third, and so on; and it is easy to see - since the two circles in the figure are tangential to the arc f D h, described from A as a centre with the radius A D - that any measurement found on one circle cannot be found on any other, and therefore that no doubt need occur as to which base a measurement applies to. When there are but two circles and bases, as in Fig. 2, since A h = A D, we have AD m+1/m-1. A d, and A H = m + 1/m - 1. AD, .•. A H = (m + 1/m - 1)2. A d.
In practice more than two circles are rarely required; but if we ha 1 n circles, and l and s were the greatest and least measurements within the range of the instruments, then l = [(m+1)/(m-1)]n.s.
(b) Suppose that the drawing is to be reduced to three-fourths, draw a straight line A B (Fig. 3) of any length. From any point A drop a perpendicular A C at right angles to A B. Lay off on AC a convenient distance AD. Then with D as a centre and radius of a length of which A D is three-fourths the original drawing, cut A B in E, join D E and the diagram is completed.
If any dimension E F from the original drawing is with compasses applied from E along E D, then F G will be the reduced dimension.
It is not necessary in practice to draw the line F G, as, by oscillating the point of the compasses, the point of perfect contact will be easily got, and will be the required reduction. The lines A B and D E may be produced in the direction of the dotted lines, thereby enabling the draughtsman to measure longer distances than that for which the diagram was at first constructed, should occasion require.