Perhaps one of the most peculiar and difficult patterns to mark out is that for a conical pipe which fits inside another conical pipe, and whose centre lines do not meet and are also inclined to each other. The position of the pipes can be best understood by reference to the plan and elevation in Fig. 348.
In this case, and in most others, the main difficulty lies in obtaining points on the curves of intersection of the two pipes; as when this is accomplished very little trouble is experienced in afterwards laying out the patterns.
To obtain points on the joint curve several methods can be used. In this case the idea is to take cutting planes which all pass through the apex p' of one cone, and thus give triangular sections; the sections of the other cone being elliptical. Where the pairs of triangles and ellipses intersect will give points on the curve of interpenetration. It will be as well to explain how to obtain one set of points. A semicircle is described on the base of the inclined cone and divided into six parts. Take the point marked 2. A line is drawn square to the base, giving a' and this point joined to p'. A projector is drawn from a' to the centre line of the cone in plan, giving a; then a 2 in the plan is made equal to a' 2 in the elevation, the points marked 2 being joined to p. The line d' e' is bisected and a horizontal line (f g) drawn through its middle point c'. On f g a semicircle is described, and a perpendicular (c' n) dropped from c' on to it; the length, c b in the plan, being made equal to c' n, and d and e obtained by projecting down from the corresponding points in the elevation. Through the points d b e b an ellipse is drawn, or such parts of it as are required to cut the lines marked p 2. Thus four points, h k l m, are found, and these projected up to the elevation. In the same manner any other number of points on the joint curve can be obtained.
To strike out the pattern for the outside cone: Lines are first drawn from t, in the plan, through each point on the curves - thus, to show two, t x and t z - and from the lengths of arcs obtained, the girth curve on the pattern is laid out and radial lines drawn, as shown by T X and T Z. The points on the pattern holes are found by running lines from the points on the elevation of the joint curve to the outside of the cone; thus, the lengths T K and T H on the pattern are respectively equal to t' k" and t' h" on the elevation. In a similar way all the other points required are obtained.
The pattern for the inclined cone or inside tube is set out in the usual way, lines being run out to the outside line of the cone and lengths taken off. Thus, P 3' and P 3" on the pattern are made equal in length to the corresponding lines p' 3' and p' 3" on the elevation
In complicated work of the above description, the setting out must be done as accurately as possible if it is desired that the parts shall fit neatly together.