This result may be verified by dropping lines vertically from the points in H I across the plan, intersecting them with the radial lines in the plan of corresponding number. Thus a line dropped from point 4 on H I should intersect the radial line M 4 at the same point (43) established upon it by measuring the distance upon line 41 from A D to A B. Having thus obtained the shape of the oblique cut as it would appear in plan, the next step is to set off upon the lines previously drawn through H1 I1 the width of the oblique cut in plan as measured upon lines of corresponding number. Therefore, with E G as a basis of measurement, with the dividers take the distance on each of the several cross lines 22, 33, 43, 53, etc., from E G to one side of the plan of the oblique cut just de-sen bed, and set off the same distance on each side of H1 T1 on the corresponding lines. A line traced through the points thus obtained will be an ellipse.
Fig. 213.- To Construct an Ellipse from. Two Circles by Intersecting Lines.
Fig. 214. - To Draw an Ellipse within a Given Rectangle, by Means of Intersecting Lines.
70. To Construct an Ellipse to Given Dimensions by the Use of Two Circles and Intersecting Lines. - In Fig. 213, let it be required to construct an ellipse, the length of which shall equal A B and the width of which shall equal H F. Draw A B and H F at right angles, intersecting at their middle points, K. From K as center, and with one-half of the length A B as radius, describe the circle A C B D. From K as center, and with one-half of the width H F as radius, describe the circle E F G H. Divide the larger circle into any convenient number of equal parts, as shown by the small figures 1, 2, 3, 4, etc. Divide the smaller circle into the same number of equal and corresponding parts, as also shown by figures. By means of the T-square, from the points in the outer circle draw vertical lines, and from points in the inner circle draw horizontal lines, as shown, producing them until they intersect the lines first drawn. A line traced through these points of intersection will be an ellipse.
71. To Draw an Ellipse within a Given Rectangle by Means of Intersecting Lines. - In Fig. 214, let E D B A be any rectangle within which it is required to construct an ellipse. Bisect the end A E, obtaining the point F, from which erect the perpendicular F G, dividing the rectangle horizontally into two equal portions. Bisect the side A B, obtaining the point H, and draw the perpendicular H I, dividing the rectangle vertically into two equal portions. The lines F G and H I are then the axes of the ellipse. F G represents the major axis, and H I the minor axis. Divide the spaces F E, F A, G D and G B into any convenient number of equal parts, as shown by the figures 1, 2, 3. From these points in F E and G D draw lines to I, and from the points in F A and G B draw lines to the point H. Divide F C and G C also into the same number of equal parts, as shown by the figures, and from H and I through each of these points draw lines, continuing them till they intersect lines of corresponding number in the other set, as indicated. A line traced through the several points of intersection between the two sets of lines, as shown in the engraving, will be an ellipse.
Besides the above methods for drawing correct ellipses there are several methods for drawing figures approximating ellipses more or less closely, but cornposed of arcs of circles, which it is sometimes necessary to substitute for true ellipses for constructive reasons. The ellipse has been described above as a curve drawn with a constantly changing radius. If, instead of using an infinite number of radii, some finite number be assumed, it will appear that the greater the number assumed the more nearly will it approach a perfect ellipse. Thus, a curve very much like an ellipse can be drawn, each quarter of which is composed of arcs drawn from two centers. If the number of centers be increased to three, the curve comes much nearer a true ellipse, and with four or five centers to each quarter, the curve thus produced can scarcely be distinguished from the perfect ellipse.
Fig. 215.-First Method.
Fig. 216. - Second Method.
To Draw an Approximate Ellipse with the Compasses, the Length only Being Given.
12. To Draw an Approximate Ellipse with the Compasses, the Length only being Given. - In Fig. 215, let A C be any length to which it is desired to draw an elliptical figure. Divide A C into four equal parts. From 3 as center, and with 3 1 as radius, strike the arc B 1 D, and from 1 as center, and the same radius, strike the arc B 3 D, intersecting the arc first struck in the points B and D. From B, through the points 1 and 3, draw the lines B E and B F indefinitely, and from D, in like manner, draw the lines D G and D H. From the point 1 as center, and with 1 A as radius, strike the arc E G, and from 3 as center, with the same radius, or its equivalent, 3 C, strike the arc H F. From D as center, with radius D G, strike the arc G II, and from B as center, with the same radius, or its equivalent, B E, strike the arc E F, thus completing the figure.
A figure of different proportions may be drawn in the same general manner as follows: Divide the length A C into four equal parts, as indicated in Fig. 216. From 2 as center, and with 2 1 as radius, strike the circle 1 E 3 F. Bisect the given length A C by the line B D, as shown, cutting the circle in the points E and F. From E, through the points 1 and 3, draw the lines E G and E H indefinitely, and from F, through the same points, draw similar lines, F I and F K. From 1 as center, and with 1 A as radius, strike the arc I A G, and from 3 as center, with equal radius, strike the arc K C H. From E as center, and with radius E G, strike the arc G D H, and from F as center, with corresponding radius, strike the arc I B K, thus completing the figure.