In Fig. 660 are shown a plan and side and end views of the head of a bathtub or the lining of a tub the body of which is constructed of wood. The end view shows the bottom corners of the tub to be rounded, as shown at C G1 and Bl F1; the plan shows the head to be semicircular, while the side view shows that the junction between the head and the sides is made on the vertical line B2 E2. It will thus be seen that the conditions here given are the same as in the previous problem - viz., an irregular flaring piece forming a transition between two quarter circles (instead of complete circles as in the previous problem) lying in planes at right angles to each other.

Fig. 660. - Elevations and Plan of Lining for a Bathtub, Showing Triangulation of the Head Piece.

Divide the quarter circle A F of the top view into any convenient number of equal spaces, as shown by the small figures. In like manner divide the quarter circle B1 F1 of the end view into the same number of equal spaces, less one, as also indicated by the small figures. From the points thus established in B1 F1 carry lines to the horizontal line B F in the top view and mark the intersection by small figures, as shown. The reason for using one less space in the quarter circle B1 F1 than in the large arc A F is because B1 F1 is not the complete profile of the end which is to be connected with A F of the top; the line F1 E being required to complete the same, thus constituting the remaining space. Having established these two sets of points in the plan, connect those of like numbers, as 1 with 1, 2 with 2, etc., by solid lines. Also connect the points in the line of the top with those of the next lower number in the base, as 2 with 1, 3 with 2, etc., as indicated by the dotted lines. These solid and dotted lines form the bases of the two sets of triangles shown in the diagrams at the right, from which the correct measurements across the pattern are to be obtained.

To construct these diagrams extend A2 E2 of the side indefinitely to the right, as shown. At any convenient points, as J and M, drop the perpendiculars J K and M N. From the points established in the quarter circle B1 F1 carry lines horizontally to the right, cutting the two perpendiculars, as shown by the small figures above K and N. From J, upon J H, set off the space J 1, equal to the line 1 1 of the plan or top view, and from the point 1 thus established draw the hypoth-enuse of the triangle, terminating in the point 1 of the line J K. In like manner set off from J, upon J H, the length 2 2 of the top view, also 3 3, 4 4 and 5 5, and from the points thus established draw lines to points of corresponding designation in the line J K, all as shown. By this means triangles have been constructed the hypothenuses of which represent measurements on the surface of the finished article, taken on lines corresponding to the solid lines of the top view.

In like manner construct the second diagram of triangles shown at the extreme right, setting off from M distances equal to the length of the dotted diagonal lines in the top view, and connecting the points thus established with points of next lower number in the line M N. Then the hypothenuses of this set of triangles will give the lengths corresponding to measurements on the dotted lines of the plan or top view.

Fig. 661. - Pattern of Head Piece Shown in. Fig. 660.

Having now obtained all the necessary measurements, the pattern may be developed as shown in Fig. 661. The central portion of the pattern will correspond to A1 C1 B1 of the end view, it being simply a flat triangular piece of metal. Therefore draw any horizontal line, as C B, equal in length to C B or C1 B1 of Fig. 660. Take the space 1 1 of the first diagrams of the triangles as radius, and from C and B, respectively, as centers, strike arcs which will intersect at A. From A as center, with 1 2 of the outer line of the plan as radius, describe a small arc. From B as center, with 1 2 of the second diagrams of triangles as radius, intersect the arc as shown, thus establishing the point 2 in the upper curve of the pattern. Then from B as center, with 1 2 of the arc B1 F1 of the end view as radius, describe another small arc, and from 2 of the upper edge of the pattern as center, with 2 2 of the first diagram of triangles as radius, intersect it as shown, thus establishing point 2 in the lower line of the pattern. Proceed in this way, using alternately the stretchout of the top of the tub, as indicated by the plan view, with the hypothenuses of the second diagram of triangles to establish the points in the upper curve of the pattern, and the stretchout of the quarter circle shown in the end view with the hypothenuses of the first diagram of triangles to establish the points in the lower line of the pattern, until the points 6 and 5, or E and F, are reached. Connect E and F by a straight line and through the points from A to E and from B to F trace lines, as shown; then A E F B will be the pattern for one of the corners of the head, a duplicate of which may be reversed and transferred to the other side of the pattern, as shown by A D G C, thus completing the entire pattern of the head in one piece.