The form of a true ellipse may be very nearly approached by means of the arcs of circles, if the centres from which those arcs are struck are located in the most desirable positions for the form of ellipse to be drawn.
Fig. 78.
Thus in Figure 78 are three ellipses whose forms were pencilled in by means of pins and a loop of twine, as already described, but which were inked in by finding four arcs of circles of a radius that would most closely approach the pencilled line; a b are the foci of all three ellipses A, B, and C; the centre for the end curves of a are at c and d, and those for its side arcs are at e and f. For B the end centres are at g and h, and the side centres at i and j. For C the end centres are at k, l, and the side centres at m and n. It will be noted that, first, all the centres for the end curves fall on the line of the length or major axis, while all those for the sides fall on the line of width or the minor axis; and, second, that as the dimensions of the ellipses increase, the centres for the arcs fall nearer to the axis of the ellipse. Now in proportion as a greater number of arcs of circles are employed to form the figure, the nearer it will approach the form of a true ellipse; but in practice it is not usual to employ more than eight, while it is obvious that not less than four can be used. When four are used they will always fall somewhere on the lines on the major and minor axis; but if eight are used, two will fall on the line of the major axis, two on the line of the minor axis, and the remaining four elsewhere.
Fig. 79.
In Figure 79 is a construction wherein four arcs are used. Draw the line a b, the major axis, and at a right angle to it the line c d, the minor axis of the figure. Now find the difference between the length of half the two axes as shown below the figure, the length of line f (from g to i) representing half the length of the figure (as from a to e), and the length or radius from g to h equalling that from e to d; hence from h to i is the difference between half the major and half the minor axis. With the radius (h i), mark from e as a centre the arcs j k, and join j k by line l. Take half the length of line l and from j as a centre mark a line on a to the arc m. Now the radius of m from e will be the radius of all the centres from which to draw the figure; hence we may draw in the circle m and draw line s, cutting the circle. Then draw line o, passing through m, and giving the centre p. From p we draw the line q, cutting the intersection of the circle with line a and giving the centre r. From r we draw line s, meeting the circle and the line c, d, giving us the centre t. From t we draw line u, passing through the centre m. These four lines o, q, s, u are prolonged past the centres, because they define what part of the curve is to be drawn from each centre: thus from centre m the curve from v to w is drawn, from centre t the curve from w to x is drawn. From centre r the curve from x to y is drawn, and from centre p the curve from y to v is drawn. It is to be noted, however, that after the point m is found, the remaining lines may be drawn very quickly, because the line o from m to p may be drawn with the triangle of 45 degrees resting on the square blade. The triangle may be turned over, set to point p and line q drawn, and by turning the triangle again the line s may be drawn from point r; finally the triangle may be again turned over and line u drawn, which renders the drawing of the circle m unnecessary.
To draw an elliptical figure whose proportion of width to breadth shall remain the same, whatever the length of the major axis may be: Take any square figure and bisect it by the line A in Figure 80. Draw, in each half of the square, the diagonals E F, G H. From P as a centre with the radius P R draw the arc S E R. With the same radius draw from O as a centre the arc T D V. With radius L C draw arc R C V, and from K as a centre draw arc S B T.