Proportional Reduction And Enlargement

The principles of reduction and enlargement are based on the properties of a triangle in which there is a progressive diminution from base-line to apex (the proportionate division of

lines described on p. 365 is founded on this principle). Any line, therefore, drawn parallel to the base of a triangle (351:1) will form another triangle either smaller or larger, of similar angles and therefore of similar shape. If a series of lines are drawn from points on the base of a triangle (351 :2A) to the apex they will cut any other parallel line (351:2B) in exactly the same proportion. The triangle can be of any shape, but in practice either equilateral or right-angled triangles are normally used.

Where the figure to be reduced (or enlarged) is a rectangle it is only necessary to draw the diagonal (351:3) to create two triangles in which the smaller rectangle (ab) is in strict proportion to the larger (AB). If a given moulding has to be reduced in section, for example to two-thirds its original size, then the procedure is shown in 351:4. The full-size section CAB is first drawn, and at a convenient distance the required height of the reduced section DE. A triangle AXC is then drawn on base-line AC to enclose DE. If DE is placed centrally as shown then the triangle will be equal sided, but if DE is placed on a horizontal line drawn from C it will be a right-angled triangle. Salient points of the moulding 1, 2, 3, 4, etc. are then drawn from the profile outline to the bed AC, and from there diminished to X, cutting DE in exactly the same proportion. The thickness of the reduced moulding section FG is then enclosed in a similar triangle erected on AB and additional salient points produced and diminished to Y, yielding points on FG which can be stepped off on the reduced section DE, FG. Alternatively, the reduced thickness can be plotted as shown in 351:5, where salient points in the thickness are drawn to the base-line CH and from centre C swung round to AC extended to G, and from there diminished to X, cutting DF extended to J. From DJ with centre F they are swung back to plot the reduced thickness of the required section FDE. In all cases the procedure can be reversed for the enlargement of any given section, and if the moulding section FDE (351:5) is to be enlarged to CAB then FDE is drawn first and the enlarged section developed from it.

The application of these methods to turned-work (balusters, table legs, etc.) is identical, and it is only necessary to draw an axis or centreline and treat one half of the turning as a flat-

bedded moulding as described above. If reeded or fluted columns in tapered legs or pilasters have to be reduced then the simpler method shown in 352:1 can be used. The thickness of the reduced section EF can be determined by describing an arc from centre C with radius BC in the section ABCD, and describing a further arc with centre F and radius taken from F to the plotted point to yield the proportionate thickness FG. If the moulding arcs are compass drawn and must also be proportionately reduced in depth (352:2), then draw in the centre-line EF for the radius centres of the arcs, and with centre D and radius DE describe an arc to DC, and from there diminish to G to give the required arc to H, which in turn will yield the compass line HJ.

Where a moulding has to be reduced in width but not in thickness, then it can be done by drawing the normal section BAC (352:3) and the reduced width AD at any angle from A. Salient lines are drawn parallel to AB as shown and swung from centre A to CA extended to D, while salient lines in the thickness which is not to be reduced are drawn horizontally. It should be pointed out that in all the foregoing examples the greater the number of salient points the more accurate will be the plotting of the required curves in the altered profile.

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