half-plan into an equal number of parts and numberthem as shown.3. With vertex A as a center and with dividers, seta distance equal to AC and draw an arc for thestretch-out of the bottom of the cone.4. Set the dividers equal to the distance of thestep-offs on the half-plan and step off twice as manyspaces on the arcs as on the half-plan; number thestep-offs 1 to 7 to 1, as shown in the illustration (fig.2-52).5. Draw lines connecting A with point 1 at eachend of the stretch-out. This arc, from 1 to 7 to 1, is equalin length to the circumference of the bottom of the cone.6. Now, using A for a center, set your dividersalong line AC to the length of AD. Scribe an arc throughboth of the lines drawn from A to 1.The area enclosed between the large and small arcsand the number 1 line is the pattern for the frustum ofa cone. Add allowance for seaming and edging andyour stretch-out is complete.TRIANGULAR DEVELOPMENTTriangulation is slower and more difficult thanparallel line or radial line development, but it is morepractical for many types of figures. Additionally, it isthe only method by which the developments of warpedsurfaces may be estimated. In development bytriangulation, the piece is divided into a series ofFigure 2-52.—Radial line development of a frustum of a cone.triangles as in radial Line development. However, thereis no one single apex for the triangles. The problembecomes one of finding the true lengths of the varyingoblique lines. This is usually done by drawing a true,length diagram.An example of layout using triangulation is thedevelopment of a transition piece.The steps in the triangulation of a warpedtransition piece joining a large, square duct and asmall, round duct are shown in figure 2-53. The stepsare as follows:1. Draw the top and front orthographic views(view A, fig. 2-53).2. Divide the circle in the top view into a numberof equal spaces and connect the division points with AD(taken from the top part of view D, fig. 2-53) from pointA. This completes one fourth of the development. Sincethe piece is symmetrical, the remainder of thedevelopment may be constructed using the lengths fromthe first part.It is difficult to keep the entire developmentperfectly symmetrical when it is built up from smalltriangles. Therefore, you may check the overallsymmetry by constructing perpendicular bisectorsof AB, BC, CD, and DA (view E, fig. 2-53) andconverging at point O. From point O, swing arcs aand b. Arc a should pass through the numberedpoints, and arc b should pass through the letteredpoints.FABRICATION OF EDGES, JOINTS,SEAMS, AND NOTCHESThere are numerous types of edges, joints, seams,and notches used to join sheet-metal work. We willdiscuss those that are most often used.EdgesEdges are formed to enhance the appearance of thework, to strengthen the piece, and to eliminate thecutting hazard of the raw edge. The kind of edge thatyou use on any job will be determined by the purpose,by the sire, and by the strength of the edge needed.The SINGLE-HEM EDGE is shown in figure2-54. This edge can be made in any width. In general,the heavier the metal, the wider the hem is made. Theallowance for the hem is equal to its width (W in fig.2-54).2-19