Interpntration

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

Interpenetration Outline

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B

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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Interpenetration Humanities

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.

Intersecting Solids Engineering DrawingAuxiliary View Engineering Drawing

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.

Interpenetration Prism

Fig. 12.12

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Cylinder Interpenetration

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

Interpenetration Prism

Fig. 12.14

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Interpenetration Prism

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drawn the circle in the plan view, it is projected up to the sides of the cone in the front elevation, and points 3 at the corners of the prism are the lowest points of contact.

A casting with a rectangular base and a circular-section shaft is given in Fig. 12.16. The machining of the radius R1 in conjunction with the milling of the flat surfaces produces the curve shown in the front elevation. Point 1 is shown projected from the end elevation. Section AA produces a circle of 0X in the plan view and cuts the face of the casting at points marked 2, which are transferred back to the section plane. Similarly, section BB gives 0Y and points marked 3. Sections can be taken until the circle in the plane view increases in size to R2; at this point, the interpenetration curve joins a horizontal line to the corner of the casting in the front elevation.

Engineering Drawing Casting

In Fig. 12.17 a circular bar of diameter D has been turned about the centre line CC and machined with a radius shown as RAD A. The resulting interpenetration curve is obtained by taking sections similar to section XX. At this section plane, a circle of radius B is projected in the front elevation and cuts the circumference of the bar at points E and F. The projection of point F along the section plane XX is one point on the curve. By taking a succession of sections, and repeating the process described, the curve can be plotted.

Note that, in all these types of problem, it rarely helps to take dozens of sections and then draw all the circles before plotting the points, as the only result is possible confusion. It is recommended that one section be taken at a time, the first roughly near the centre of any curve, and others sufficiently far apart for clarity but near enough to maintain accuracy. More sections are generally required where curves suddenly change direction.

Interpenetration Pencil Drawings

Fig. 12.17

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