A cone is a pyramid with a circular base (345:1), and a vertical cut through the axis will yield a section which is a triangle, parallel to the axis a hyperbola, and parallel to one of the sides a parabola. If the cut is parallel with the base then the section will be a circle (345:2), and inclined to the base an ellipse. Of these conic sections the ellipse is of considerable importance in the setting out of oval-shaped work, gate-leg tables, etc., and although it can be drawn by hand it is more satisfactory to construct it geometrically.


Several methods are available of which the foci method (345:3) is the most common. The major axis which is to be the larger diameter of the ellipse is drawn first, and the minor axis at right angles to it. Focal points (foci) are then found by taking half the length of the major axis, and from point A on the minor axis striking an arc to cut the major axis at the foci Fl and F2. If pins




345 Conic sections, ellipses, etc.


are inserted at A, Fl and F2 and a length of string stretched taut round them, with pin A removed and replaced by a pencil the curve can be traced by keeping the pencil taut against the string. Another familiar method (354:4) uses a trammel composed of a thin lath or strip of stout paper. The major and minor axes are drawn as before, the trammel is then marked with half the length of the major axis AB and half the minor axis BC, and if the point C is kept on the former and the point A on the latter then the point B will trace out the ellipse.

Ellipses can also be constructed by the rectangle method (345:5), in which both major and minor axes are divided into the same number of equal parts, and lines drawn through each division will yield points through which the




ellipse can be drawn. Where, however, a geometric ellipse gives too elongated a shape, an approximate ellipse of four arcs correspond­ing to the isometric projection of a circle can be used as shown in 345:6. First construct a rhombus whose sides are equal to the diameter of the circle which is to be the basis of the ellipse. Draw in the diagonals and from point A bisect two sides of the rhombus at E, F. With radius CE and centres C and D describe the smaller arcs, and with centres A and B the larger arcs.




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Compound shapes cannot be measured accurately from a two-dimensional drawing of elevation and plan, and require development, i. e. the parts opened out and flattened on one plane. Figure 346:1, 2 illustrates a typical hopper with all sides splayed, from which it will be seen that the elevational outline is correct, but the given width of both sides and ends do not allow for the actual splay or leaning outwards towards the observer which cannot be shown. To develop these true measurements first draw the plan ABCD (346:3), extend as shown by the dotted lines, draw AB, BC, CD and DA parallel to the plan at a distance equal to X and Y in 346:1, 2 and complete.

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