In Fig. 572, A B B' A' of the plan shows the rectangular base and C E D F the elliptical top of an article, the sides of which are required to form a transition between the two outlines. A" C D' B" is an end elevation of the same, showing its vertical hight X Y. An inspection of the plan will show that the article consists of four symmetrical quarters, and that that part of either quarter lying between the curved outline of the top and the extreme angle of the base, as the part E D B, is a portion of the envelope of an oblique elliptical cone, of which E D is the base and B the apex.

Fig. 572. - Plan and Elevation of Transition Piece.

Fig. 573. - Pattern for One-Quarter of Article Shown in Fig. 572.

The conditions here given are exactly the same as in the two preceding problems; a different method of obtaining the pattern has, however, been employed, not because it is better but for the sake of variety, leaving the reader to judge which method is the more available in any given case. Divide E D into any convenient number of equal spaces, as shown by the small figures, and from the points thus obtained draw lines to B. These lines will form the bases of a series of triangles whose common altitude is equal to the hight of the article, X Y, and whose hypothenuses when obtained will be the real distances from B in the base to the points assumed in the curve of the top. To construct such a diagram of triangles, first draw any line, as L M, and from M lay off the distances shown by solid lines in plan, thus making M 1 equal to B 1, M 2 equal to B 2, etc. At right angles to L M draw M N, in hight equal to the straight hight of the article, as shown by X Y of elevation, and connect the points in M L with N. Also setoff the distance D d from M, and draw N d. If E d was different in length from D d, this distance would be set off from M and a line drawn to N.

To develop the pattern first draw any line, as E B of Fig. 573, equal in length to N 1 of the diagram. With B of pattern as center, and N 2 of the diagram of triangles as radius, describe a small arc, which intersect with another arc struck from E of pattern as center and E 2 of plan as radius, thus establishing point 2 of pattern. Proceed in this manner, using the distance between points in plan for the distance between similar points in pattern, and the hypothenuses of the triangles in the diagram in Fig. 572 for the distances to be set off from B of pattern on lines of similar number. Through the points thus obtained trace a line, as shown by E D. With B of pattern as center, and B d of plan as radius, strike a small arc, which intersect with another struck from D of patterns as center, and N d of diagram as radius, thus establishing the point d of the pattern. Draw D d and d B. With B of the pattern as center, and B e of the plan as radius, strike a small arc, which intersect with another struck from B as center, with a radius equal to N d of the diagram of triangles. Draw E e and e-B; then E e B d D will be the pattern for one-quarter of the article.

In performing the work of development of the pattern it will be found convenient as well as more accurate to use two pairs of compasses, one of which should remain set to the space used in dividing the curve E D of the plan, while the other may be changed to the varying lengths of the hypothenuses in the diagram of triangles.