To find the shape, let A (Fig. 99) be the plan, and B the section, of a given dome. From a draw a c at right angles to a b; find the stretch-out (Art. 524) of 0 b, and make dc equal to it; divide the arc o b and the line d c each into a like number of equal parts, as 5 (a large number will insure greater accuracy than a small one); upon c, through the several points of division in cd, describe the arcs 0 do, 1 e 1, 2 f 2, etc.; make do equal to half the width of one of the boards, and draw 0 s parallel to a c; join s and a, and from the points of division in the arc ob drop perpendiculars, meeting a s in i j k l; from these points draw i 4,j 3, etc., parallel to ac; make do, e I, etc., on the lower side of a c, equal to d o, e I, etc., on the upper side; trace a curve through the points 0, 1,2, 3, 4, c, on each side of d c; then 0 c 0 will be the proper shape for the board. By dividing the circumference of the base A into equal parts, and making the bottom, 0 d 0, of the board of a size equal to one of those parts, every board may be made of the same size. In the same manner as the above, the shape of the covering for sections of another form may be found, such as an ogee, cove, etc.
To find the curve of the boards when laid in horizontal courses, let A B C (Fig. 100) be the section of a given dome, and DB its axis. Divide B C into as many parts as there are to be courses of boards, in the points 1, 2, 3, etc.; through 1 and 2 draw a line to meet the axis extended at a; then a will be the centre for describing the edges of the board F. Through 3 and 2 draw 3 b; then b will be the centre for describing F. Through 4 and 3 draw 4d; then d will be the centre for G. B is the centre for the arc 1 0. If this method is taken to find the centres for the boards at the base of the dome, they would occur so distant as to make it impracticable; the following method is preferable for this purpose: G being the last board obtained by the above method, extend the curve of its inner edge until it meets the axis, D B, in e; from 3, through e, draw 3 f, meeting the arc A B in f; join f and 4,f and 5, and f and 6, cutting the axis, D B, in s, n, and m; from 4, 5, and 6 draw lines parallel to A C and cutting the axis in c, p, and r; make c 4 (Fig. 101) equal to c 4 in the previous figure, and c s equal to c s also in the previous figure; then describe the inner edge of the board H, according to Art. 516; the outer edge can be obtained by gauging from the inner edge. In like manner proceed to obtain the next board - taking p 5 for half the chord, and p n for the height of the segment. Should the segment be too large to be described easily, reduce it by finding intermediate points in the curve, as at Art. 515.
241. - Polygonal Dome: Form of Angle-Rib. - To obtain the shape of this rib, let A G H (Fig. 102) be the plan of a given dome, and C D a vertical section taken at the line ef. From 1, 2, 3, etc., in the arc CD draw ordinates, parallel to A D, to meet fG; from the points of intersection on fG draw ordinates at right angles to fG; make s 1 equal to 0 1, s 2 equal to 0 2, etc.; then GfB, obtained in this way, will be the angle-rib required. The best position for the sheathing-boards for a dome of this kind is horizontal, but if they are required to be bent from the base to the vertex, their shape may be found in a similar manner to that shown at Fig. 99.