In Fig. 644, let A B C D be the elevation and E F G H half the plan of a bathtub. An inspection of the drawing shows that neither the segments forming the head nor those of the foot of the tub are concentric, and that their upper and lower bases are not parallel. Therefore the figures which they constitute are irregular in character, and the only available method by which the various dimensions and curves constituting the patterns of the same can be ascertained is by dividing their surfaces into small triangles, which can most easily be accomplished in the following manner: Divide each of the curves J 1 and G H, forming the plan of the head piece, into the same number of equal parts, numbering each the same, as shown, and connect points of similar number by solid lines. Also connect each point in J I, the line of the bottom, with the point of next higher number in G H, the line of the top, by a dotted line, all as shown. The curves E F and L K, forming the plan of the foot piece, arc also to be divided into spaces and the points connected by solid and dotted lines in the same manner as those of the head.

Fig. 644. - Plan and Elevation of Bathtub, with Diagrams of Triangles, Section of Top and Pattern of Side Piece.

The solid and dotted lines thus drawn between the points in the two curves of the plan will form the bases of a series of right-angled triangles, whose hypoth-enuses (after the altitudes are obtained) will give the real distance between the points whose number they bear upon the finished article. As, owing to the slant of the top line A B of the elevation, the triangles will be of differing hights, the simplest way of constructing them will be as follows: Upon D C of the elevation extended, as a base, erect a perpendicular line, M N From N on the base line set off the various lengths of the solid lines in the plan of the head prece, as shown toward Q. From each of the points in the curve G H erect perpendicular lines, cutting A B of the elevation; and from these points of intersection carry lines horizontally to the right, cutting the line M N, numbering each point to correspond with the points in G H, all as shown. Lines connecting points of similar number at M and Q will be the hypothenuses required, or the real distances between points of similar number in the top and bottom of the finished article. In a similar manner erect another perpendicular, O P, and set off from P on the base line the lengths of the several dotted lines in the plan of the head piece, as shown toward R. The bights of the points in the curve of the top can be determined upon the line O P by continuing the lines drawn from A B toward M till they intersect O P. Each point in the base P R is now to be connected with the point of the next higher number in O P by a dotted line. The various hypothenuses drawn between O and R will then be the correct distances between the points connected by the dotted lines of the plan. (The numbers in P R correspond with those in I J of the plan and not with H G). The diagrams from which the dimensions for the foot piece are obtained arc shown at S T U and V W X at the left of the elevation and are obtained in a manner exactly similar to those just obtained for the head piece, all of which is clearly shown by the lines of the drawing.

Fig. 645. - Half Pattern of Head Piece.

Fig. 646. - Half Pattern of Foot Piece.

From an inspection of the drawing it will be seen that the line E F G H does not represent the true lengths or measurements taken on the top line of the tub, because A B, not being horizontal, is longer than the line E H, its equivalent in the plan, and therefore a true section on the line A B must be obtained, as shown above the elevation. This may be accomplished in the following manner: At any convenient distance above A B draw E1 H1 parallel to A B, and from all of the points previously obtained on A B carry lines at right angles to A B, cutting E1 H1, and extend them beyond indefinitely, numbering each line to correspond with the point in E F G H from which it is derived. On the line 2 set off from E1 H1 a distance equal to the distance of point 2 of the plan from the line E H as measured on the vertical line; on line 3 set off as before a distance equal to the distance of point 3 of the plan from E H, and so continue until the distances from E H of all the points in E F G H have been transferred in like manner to the new section. Then a line traced through these points, as shown by E1 F1 G1 H1, will be a section or plan on the line A B, from which measurements can be taken in developing the upper edge of the pattern.

Having obtained, by means of the various diagrams constructed in connection with the elevation, the correct distances between all the points originally assumed in the plan, the pattern may now be developed by simply reproducing all of these distances or measurements in the order in which they occur upon the plan. For the pattern of the, head piece assume any line, as I H of Fig. 645, which make equal in length to C B of the elevation, or what is the same thing, equal to 1 1 of the diagram M N Q. From H as a center, with a radius equal to 1 2 of the section of top, strike a small arc, which intersect with another arc struck from I as a center, with a radius equal to 1 2 of the diagram of dotted lines O P R, thus establishing tin-point 2' of the pattern. From 2' as center, with a radius equal to 2 2 of the diagram of solid lines M N Q, strike a small are, which intersect with another struck from point I as a center, with a radius equal to 1 2 of the line I J of the plan, thus establishing the point 2 of the pattern. Continue this operation, using in numerical order the distances taken from the top section, in connection with the distances obtained from the diagram of dotted lines O P R, to form the top line of the pattern, and the distances taken from the diagram of solid lines M N O, in connection with the distances measured upon the bottom line I J of the plan, to form the bottom line of the pattern, all as indicated by the solid and dotted lines drawn across a portion of the pattern. Then I H G J will be one-half the pattern of the head piece. The pattern for the foot piece is developed in exactly the same manner by making E L equal to A D of the elevation, and using the diagram of dotted lines V W X to measure upon the pattern the distances indicated by the dotted lines upon the plan, and the diagram of solid lines S T U to measure upon the pattern the distances indicated by the solid lines across the plan, the distances forming the top line of the pattern being taken from E1 F1 of the top section while the distances forming the bottom line of the pattern sure taken from the line L K of the plan.

The pattern for the Bat portion of the side F G J K can be obtained as follows: Parallel to K J of the plan draw any line, as K1 J1. At right angles to K J of the plan project lines from points K. J, F and G, cutting K1 J1, as shown, establishing the points K1 and J1, and continuing the lines from points F and G indefinitely. From K1 of the pattern as a center, with a radius equal to 8 8 of the diagram S T U. or of Fig. 644, strike a small are, cutting the line projected from point F of the plan, as shown at F2 of the pattern. From J1 of the pattern as a center, with a radius equal to 7 7 of the diagram M N Q, or of Fig. 644. strike a small arc, cutting the line projected from the point G of the plan, as shown at G2 Draw the lines K1 F2, F2 G2 and G2 J1; then K1 F2G2 J1 will be the pattern of the flat portion of the side.

The patterns of the several parts can be joined together, by any convenient method of duplication, in such a manner as to produce as much of the entire pattern in one piece as it is desired.