Many objects are formed by a collection of geometrical shapes such as cubes, cones, spheres, cylinders, prisms, pyramids, etc., and where any two of these shapes meet, some sort of curve of intersection or interpenetration results. It is necessary to be able to draw these curves to complete drawings in orthographic projection or to draw patterns and developments.
The following drawings show some of the most commonly found examples of interpenetration. Basically, most curves are constructed by taking sections through the intersecting shapes, and, to keep construction lines to a minimum and hence avoid confusion, only one or two sections have been taken in arbitrary positions to show the principle involved; further similar parallel sections are then required to establish the line of the curve in its complete form. Where centre lines are offset, hidden curves will not be the same as curves directly facing the draughtsman, but the draughting principle of taking sections in the manner indicated on either side of the centre lines of the shapes involved will certainly be the same.
If two cylinders, or a cone and a cylinder, or two cones intersect each other at any angle, and the curved surfaces of both solids enclose the same sphere, then the outline of the intersection in each case will be an ellipse. In the illustrations given in Fig. 12.7 the centre
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lines of the two solids intersect at point O, and a true view along the line AB will produce an ellipse.
When cylinders of equal diameter intersect as shown in Fig. 12.8 the line at the intersection is straight and at 45°.
  
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Fig. 12.10
Figure 12.9 shows a branch cylinder square with the axis of the vertical cylinder but reduced in size. A section through any cylinder parallel with the axis produces a rectangle, in this case of width Y in the branch and width X in the vertical cylinder. Note that interpenetration occurs at points marked 3, and these points lie on a curve. The projection of the branch cylinder along the horizontal centre line gives the points marked 1, and along the vertical centre line gives the points marked 2.
Figure 12.10 shows a cylinder with a branch on the same vertical centre line but inclined at an angle. Instead of an end elevation, the position of section AA is shown on a part auxiliary view of the branch. The construction is otherwise the same as that for Fig. 12.9.
In Fig. 12.11 the branch is offset, but the construction is similar to that shown in Fig. 12.10.
Figure 12.12 shows the branch offset but square with the vertical axis.
Figure 12.13 shows a cone passing through a cylinder. A horizontal section AA through the cone will give a circle of 0P, and through the cylinder will give a rectangle of width X. The points of intersection of the circle and part of the rectangle in the plan view are projected up to the section plane in the front elevation.
Fig. 12.11
Figure 12.12 shows the branch offset but square with the vertical axis.
Figure 12.13 shows a cone passing through a cylinder. A horizontal section AA through the cone will give a circle of 0P, and through the cylinder will give a rectangle of width X. The points of intersection of the circle and part of the rectangle in the plan view are projected up to the section plane in the front elevation.
Fig. 12.12
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Fig. 12.15 The plotting of more points from more sections will give the interpenetration curves shown in the front elevation and the plan. Figure 12.14 shows a cylinder passing through a cone. The construction shown is the same as for Fig. 12.13 in principle. Figure 12.15 shows a cone and a square prism where interpenetration starts along the horizontal section BB at point 1 on the smallest diameter circle to touch the prism. Section AA is an arbitrary section where the projected diameter of the cone 0X cuts the prism in the plan view at the points marked 2. These points are then projected back to the section plane in the front elevation and lie on the curve required. The circle at section CC is the largest circle which will touch the prism across the diagonals in the plan view. Having Fig. 12.14
