In Fig. 738, A B C is the elevation of the inner curve A' H C' of the plan and EBD that of the outer curve E' F' D'. As will be seen by inspection, the outer curve is drawn from G as center, that portion of the opening from the springing line down to the springing line of the inner curve, as E 12, being straight and vertical. The pattern of the soffit could have been obtained in exactly the same manner had the elevation of this outer curve been a semi-ellipse.
Fig. 738. - Plan, Elevation and Extended Sections of an Arch in a Circular Wall.
Divide A B of the elevation line into any convenient number of equal parts, and with the T-square parallel with center line B H drop lines from the points in A B, cutting A' H of plan, as shown by the small figures 1 to 6. As the semicircle representing the outer curve of wall is struck from G as center, divide 12 B into the same number of parts as was A B. and drop lines from these points to E' F' of plan, as shown. Connect opposite points in E' F' of plan with those in A' H. as indicated by the solid lines. Also connect the points of the plan obliquely, as shown by the dotted lines, thus dividing the plan of the soffit of the arch into triangles. In order to ascertain the true distances which these lines drawn across the plan represent it will be necessary to construct a series of sections of which they arc the bases, as shown in Figs. 739 and 740.
In Fig. 739 is shown a diagram of sections corresponding to the solid lines in plan, to construct which proceed as follows: Draw the right angle P Q R, and, measuring from Q, set off on Q P the length of lines dropped from points in A B of elevation to A F, as shown by the figures 2 to 6. Likewise set off from and connect the points thus obtained with the points in P Q, as indicated by the' numbers in the plan. Thus conned 6 with 7, 5 with 8, 4 with 9, etc.
Q on Q R the length of solid lines in plan, as shown by the small figures 7 to 12. With the T-square parallel with P Q, erect lines from the points in Q R, and. measuring from Q R. set off on these lines the length of lines of corresponding number in E B F of elevation, the Dotted Lines of the Plan.
Fig. 739. - Diagram of Sections on the Solid Lines of the Plan.
Fig. 740. - Diagram of Sections on
To construct a diagram of sections corresponding to the dotted lines of the plan draw the right angle S T U. Fig. 740, and set off on ST the length of lines dropped from points in A B of elevation to A F. or tranfer the distances in P Q, Fig. 739. On T U set off the length of dotted lines in plan, and from the points thus obtained draw lines parallel with S T, making these lines of the same length as those dropped from points in E B of elevation to E F. Connect the upper ends of lines 8 to 12 with points in S T, as indi-cated by the dotted lines in plan. Thus connect 2 with 12, 3 with 11, 4 with 10, etc.
The next step is to obtain the distance between points in A B and F B of elevation as if measured upon the curved surfaces of the wall. It is therefore necessary to develop extended sections of the two curves of the arch as shown at the left in the engraving. The development of the curve of the inner side of the arch is projected directly from the elevation in the following manner: On A F extended lav off J K, equal to the curved line A' H of plan, making the straight line equal to the stretchout of half of the curve of the plan, as indicated by the small figures. At right angles to J K, and from the points in the same, erect lines, making them the same length as lines of similar Dumber dropped from points in A B, or, with the T-square parallel with J K and A F. carry lines from the points in A B intersecting the vertical lines of similar num-ber, A line traced through the points of intersection, as shown by J L, Will be the desired shape. The shape on E' F' of the plan, corresponding to E B F of elevation, is obtained in a similar manner. On M N in Fig. 738 set off the stretchout of E' F' of plan, as indicated by the small figures, from the points of which erect vertical lines. On these lines set off the same length as the lilies of similar number in E B F of elevation. A line traced through the points thus obtained will give the desired outer curve of the arch.
To develop the pattern from the several sections obtained proceed in the following manner: Draw the line a e of Fig. 741, in length equal to Q R of Fig. 739, and with e as center, and M 12 of the curve M O as radius, strike a small are, 12, which intersect with one struck from a as center, and Q 12 of Fig. 739 as radius, thus establishing the point 12 of pattern. With a of pattern as center, and J 2 in the curve J L as radius, describe a small arc, 2, which intersect with one struck from point 12 of pattern as center, and 2 12 of Fig. 740 as radius, thus establishing the point 2 of pattern. With 2 of pattern as center, and 2 11 of Fig. 739 as radius, strike another small arc, which intersect with one struck from point 12 of pattern as center, and 12 11 in the curve M O as radius, thus establishing the point 11 of pattern. Continue in this way, using the tops of the sections in Figs. 739 and 740 for measurements across the pattern, and the spaces in the inner and outer curves as developed in J L and M O, Fig. 738, for the distances about the edges of the pattern, establishing the several points, as shown, through which draw the lines e f, a h and f h, thus completing the pattern for one-half the soffit of the arch. The other half of pattern can be obtained by the same method or by duplication.
Fig. 741. - Half Pattern of Soffit.