Surfaces of double curvature (domes, etc.) cannot be developed with complete accuracy, nevertheless the method illustrated in 347 is sufficiently accurate for most practical purposes. Assuming that a hemisphere has to be developed to provide the exact shapes of the constructional ribs and of the covering veneers, first draw the elevation and divide horizontally into any number of zones A, B, C, D, E, F, etc. The outer circumference of the dome is then drawn on plan, and the zones in elevation pro­jected downwards to the corresponding circles on the plan B, C, D, E, and F. The circle on plan is then divided into any number of equal parts 1, 2, 3, 4, 5, etc., and the radii drawn in. To develop any one zone in the elevation, for instance the zone contained within the horizontal lines C and D shaded on the drawing, 346 Developments (1)

draw a straight line CD through C and D from point O to meet the central axis R extended, and with centre R and radii RD and RC describe arcs. These arcs are then plotted off to correspond with the division of the appropriate circles D and C on plan and numbered similarly (the drawing shows part of the development only and 12 divisions will be required to plot the exact circumferential length of the zone).

To plot the shapes of the individual gores or lunes draw verticals from the points of inter­section of each individual circle on plan with any two adjacent radii, and where these verticals intersect the horizontal limits of each zone will lie points through which smooth curves will define the gore. The true approximate shape of

347 Developments (2)

the gore can be developed by bisecting one sector on plan 2-3, plotting off the distances from A to F in elevation along the bisecting line 2-3, and with radii equal to the corresponding circles on plan describing arcs through the plotted points. If parallel lines (radiates) to the bisecting line are then produced from the appropriate arcs of the sector to contact the arcs already drawn in the development, the intersections will provide points through which smooth curves can be drawn to yield the outline of the gore. As already mentioned the development can only be approximate because the distances measured off to provide the arcs A, B, C, D, E, Fin the development are in reality chords of the true arcs of each zone, therefore, in cutting veneers for gores, an allowance must be made for trimming in position.

GEOMETRIC DECORATION Volutes and spirals

Classical architecture made considerable use of volutes and spirals as decorative elements. The former conforms to the convolution or twist of the spiral shell of certain molluscs (snails, etc.), and 348:1 gives the method of construction in 2

1 2 3 4 5 « 7 ft » 10 11 12

348 Volutes and spirals, etc.

which a circle is first drawn and divided into any number of equal parts, in this case 1 to 12. A tangent is then drawn at O equal in length to the circumference of the circle and similarly divided. If further tangents are drawn at each point of the circumference progressively increasing in length according to the number of divisions on the first tangent, i. e. from point 1 one part, point 2 two parts, etc., they will yield points for the free-hand drawing of the curve.

Spirals are shown in 348:2, 3. In 2 the containing circle is divided into any number of equal sectors (eight are shown). One radius (radius vector) is then divided into the same number of equal parts, and concentric circles described through each division. The spiral is then unwound from point O at the centre of the circle, gaining one division as it travels through each sector. If necessary the number of divisions in the radius vector can be doubled, i. e. 16, and the curve will then travel twice round the containing circle to form a double spiral; while the spiral need not start from the centre of the circle but from any point on the radius sector provided the divisions 0,1,2,3,4, 5, etc., are marked off from the commencement of the spiral and not from the centre of the containing circle.

Spirals of constant pitch built up of quadrants are constructed as shown in 348:3. A square 1234 is first drawn and a perpendicular X erected equal in height to eight times the side of the square. The arcs are then drawn from centre 1 with radius one part (one side of the square), centre 2, two parts, centre 3, three parts, centre 4, four parts, centre 5, five parts, etc.

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