View A shows the lateral, or curved, surface of acylindrically shaped object, such as a tin can. It isdevelopable since it has a single-curved surface of oneconstant radius. The width of the development is equalto the height of the cylinder, and the length of thedevelopment is equal to the circumference of thecylinder plus the seam allowance.View B shows the development of a cylinder withthe top truncated at a 45-degree angle (one half of atwo-piece 90-degree elbow). Points of intersection areestablished to give the curved shape on thedevelopment. These points are derived from theintersection of a length location, representing a certaindistance around the circumference from a starting point,and the height location at that same point on thecircumference. The closer the points of intersection areto one another, the greater the accuracy of thedevelopment. An irregular curve is used to connect thepoints of intersection.View C, shows the development of the surface ofa cylinder with both the top and bottom truncated atan angle of 22.5° (the center part of a three-pieceelbow). It is normal practice in sheet metal work toplace the seam on the shortest side. In thedevelopment of elbows, however, the practice wouldresult in considerable waste of material, as shown inview A. To avoid this waste and to simplify cuttingthe pieces, the seams are alternately placed 180° apart,as shown in figure 8-13, view B, for a two-pieceelbow, and view C for a three-piece elbow.RADIAL-LINE DEVELOPMENT OFCONICAL SURFACESThe surface of a cone is developable because a thinsheet of flexible material can be wrapped smoothlyabout it. The two dimensions necessary to make thedevelopment of the surface are the slant height of thecone and the circumference of its base. For a rightcircular cone (symmetrical about the vertical axis), thedeveloped shape is a sector of a circle. The radius forthis sector is the slant height of the cone, and the lengtharound the perimeter of the sector is equal to thecircumference of the base. The proportion of the heightto the base diameter determines the size of the sector, asshown in figure 8-14, view A.The next three subjects deal with the developmentof a regular cone, a truncated cone, and an obliquecone.Regular ConeIn figure 8-14, view B, the top view is divided intoan equal number of divisions, in this case 12. Thechordal distance between these points is used to step offthe length of arc on the development. The radius for thedevelopment is seen as the slant height in the front view.If a cone is truncated at an angle to the base, the insideshape on the development no longer has a constantradius; it is an ellipse that must be plotted by establishingpoints of intersection. The divisions made on the topview are projected down to the base of the cone in thefront view. Element lines are drawn from these pointsto the apex of the cone. These element lines are seen intheir true length only when the viewer is looking at rightangles to them. Thus the points at which they cross thetruncation line must be carried across, parallel to thebase, to the outside element line, which is seen in its truelength. The development is first made to represent thecomplete surface of the cone. Element lines are drawnfrom the step-off points about the circumference to thecenter point. True-length settings for each element lineare taken for the front view and marked off on thecorresponding element lines in the development. Anirregular curve is used to connect these points ofintersection, giving the proper inside shape.Truncated ConeThe development of a frustum of a cone is thedevelopment of a full cone less the development of thepart removed, as shown in figure 8-15. Note that, at alltimes, the radius setting, either R_{1} or R_{2}, is a slant height,a distance taken on the surface of the cones.When the top of a cone is truncated at an angle tothe base, the top surface will not be seen as a true circle.This shape must be plotted by established points ofintersection. True radius settings for each element lineare taken from the front view and marked off on thecorresponding element line in the top view. Thesepoints are connected with an irregular curve to give thecorrect oval shape for the top surface. If thedevelopment of the sloping top surface is required, anauxiliary view of this surface shows its true shape.Oblique ConeAn oblique cone is generally developed by thetriangulation method. Look at figure 8-16 as you readthis explanation. The base of the cone is divided into anequal number of divisions, and elements 0-1, 0-2, andso on are drawn in the top view, projected down, anddrawn in the front view. The true lengths of the elements8-11