Ellipses and Ovals.—It is not necessary to measure the centering points (F) at certain specified distances from the intersection of the horizontal and vertical lines. We may take any point along the major axis, as shown, for instance, in Fig. 149. Let B be this point, taken at random. Then describe the half circle (C). We may, also, arbitrarily, take any point, as, for instance, D on the minor axis E, and by drawing the diagonal lines (F) we find marks on the circle (C), which are the meeting lines for the large curve (H), with the small curve (C). In this case we have formed an ovate or an oval form. Experience will soon make perfect in following out these directions.
Focal Points.—The focal point of a circle is its center, and is called the focus. But an ellipse has two focal points, called foci, represented by F, F in Fig. 148, and by B, B in Fig. 149.
A produced line is one which extends out beyond the marking point. Thus in Fig. 148 that part of the line K between F and G represents the produced portion of line K.
Fig. 149.
Spirals.—There is no more difficult figure to make with a bow or a circle pen than a spiral. In Fig. 150 a horizontal and a vertical line (A, B), respectively, are drawn, and at their intersection a small circle (C) is formed. This now provides for four centering points for the circle pen, on the two lines (A, B). Intermediate these points indicate a second set of marks halfway between the marks on the lines. If you will now set the point of the compass at, say, the mark 3, and the pencil point of the compass at D, and make a curved mark one-eighth of the way around, say, to the radial line (E), then put the point of the compass to 4, and extend the pencil point of the compass so it coincides with the curved line just drawn, and then again make another curve, one-eighth of a complete circle, and so on around the entire circle of marking points, successively, you will produce a spiral, which, although not absolutely accurate, is the nearest approach with a circle pen. To make this neatly requires care and patience.
Fig. 150.