All the setting out necessary to obtain the pattern construction lines can be done with a one-eighth plan and a half-section, as shown in Fig. 307. On the plan, b 0' and 0' 2' each equal half the diameter of the base of the pyramid. The centre, b, is joined to 2' and produced outwards to meet the cone base in 2, the arc, 2 0, then being bisected and the middle point, 1, joined to b. The points 2' and 1' are now swung up, around b, on to the base line, giving the points 1" and 2". These latter are then joined to c, and where the connecting lines cut the outside of cone in 0°, 1°, and 2° will determine the lengths required for the two patterns.
In marking the pattern out for the tapered square pipe, the compasses are put in centre c and opened to the point 2", the outer arc then being described to this radius. The compasses are next set to the length of the side of the pyramid base, that is, twice the length of the line 0' 2' on the plan, and five lengths to this stepped around the arc on the pattern, thus marking the points 2r. The five chords are next drawn by joining the points marked 2', the two end chords being bisected in the points 0'. It will thus be seen that there are three full sides and two halves to make up the complete pattern of four sides. The points 1' and 0' on the pattern are fixed by making the lengths of 2' 1' and 2' 0' the same as these lines on the plan. From each point radial lines are drawn to c, these being cut, to give points on the pattern curve, by drawing arcs around from the points 0°, 1°, and 2°. Thus, to give one instance, where the arc drawn from 1° intersects the radial lines 1' c, will give points on the curve of the pattern cut. These are then all joined up with even curves, as shown. The cut to form the small end of the pipe is set out by producing the top line in the elevation outwards to meet c 2" in d; then, with c d as radius, the arc is swept around, and where this intersects the lines c 2', in e, on the pattern, will give the end of the top lines; these being then drawn in, as shown, by the lines marked e e.
The outer dotted pattern, it might be useful to remember, will, if bent into shape, give the portion of the tapered square pipe which fits inside the conical dome.
For the conical dome pattern the compasses are set to the radius a 0 in the elevation, and the circular arc marked out; sixteen distances being stepped around this, each equal in length to either of the arcs 0 1 or 1 2 in the plan. From each division point radial lines are drawn to the centre A; these are then cut by arcs drawn to the respective radii, a 0°, a 1°, a 2°, from the half-section. Thus the length A 1° on the pattern equals a 1° on the section, and so on for the other lines. The points being joined with even curves, the pattern is now complete.
It is interesting to notice that the inner pattern, marked off by the dotted line A 0°, will, when bent into shape, give the portion on the cone fitting inside the tapered square pipe.
If the centre lines of the pyramid and cone do not coincide, the patterns for the parts required can still be marked out by the method shown above, the only difference being in the plan, this, perhaps, requiring to be a half or a full-size plan, according to the position of the tapered square pipe on the dome.