Prob. 11, Fig. 33. To draw a circle within a given triangle, as abc.-Bisect any two of the angles as in Prob. 2. Where the bisecting lines cross, will be the center of the required circle. In a similar manner a center may be found from which to draw a circle tangent to any three given straight lines. Prob. 12, Fig. 34. To find the foci of an ellipse.Draw the long and the short diameters of the ellipse, ab and cd, and with a radius equal to half of the long diameter, and with a center at c or d FIG. 34.-Finding the Foci of FIG. 35.-Simplified Method an Ellipse. of Drawing an Ellipse. draw the arcs e and f. Where these arcs intersect the long diameter will be the required foci. Prob. 13, Fig. 35. To draw an ellipse with a pencil and thread.—Having found the foci of the ellipse, stick a pin firmly into each focus, and looping a thread around them, allow it to be slack enough so that the pencil will draw it out to the end of the short diameter. The thread will then guide the pencil so that it will draw an ellipse. A groove should be cut around the pencil lead to prevent the thread from slipping off. Prob. 14, Fig. 36. To draw an ellipse with a trammel.--Lay out the long and the short diameters of the ellipse, ab and cd, and on a strip of paper, A, mark off 1-3 equal to half of the long diam eter, and 2-3 equal to half of the short diameter. Then, keeping point 1 on the short diameter, and point 2 on the long diameter, mark off any desired number of points at 3. A curved line passing through these points will be the required ellipse. The ellipsograph, an instrument for drawing ellipses, is made on this principle, points at 1 and 2 traveling in grooves which coincide with ab and cd. Prob. 15, Fig. 37. To draw an ellipse by tangent lines.-Make ab equal to one-half of the long di 3 a d FIG. 36. Another Method of FIG. 37.-Drawing an Ellipse Drawing an Ellipse. by Tangents. ameter of the required ellipse, and be equal to onehalf its short diameter. Divide ab and be into the same number of equal parts, and, numbering them as indicated, connect 1 and 1', 2 and 2' and so forth. A curved line starting at a, tangent to these lines, and ending at c, is one-quarter of the required ellipse. Prob. 16, Fig. 38. To draw an approximate ellipse with compasses, using four centers.-Lay out the long diameter ab, and the short diameter cd, crossing each other centrally at o. From b measure off be equal to co, one-half of the short diameter. The length ae will then be the radius gh for forming the part hk of the ellipse. From e mark off the point f, making ef equal to one half of oe. The point ƒ will be the center, and fb the radius for forming the end of the ellipse. Lines drawn from the centers g through the points ƒ determine the points at which the different curves meet. This method is not considered applicable when the short diameter is less than two-thirds of the long diameter. Prob. 17, Figs. 39 and 39a. To draw an approximate ellipse with compasses, using eight centers.Lay out the long diameter ab, and the short diameter cd crossing each other centrally at f. Construct the parallelogram aecf, and draw the diagonal ac. From e draw a line at right angles to ac, crossing the long diameter at h, and meeting the short diameter, extended, at g. Point g is the center from which to strike the sides of the ellipse, and h will be the center, subject to certain modifications for narrow ellipses, from which to strike the ends of the ellipse. To get the radius of the third curve for connecting the side and end curves, lay off a base line ab, Fig. 39A, of any convenient length, and divide it into five equal parts by the points 1, 2, 3 and 4. At one end of the line erect the perpendicular ac, equal to the end radius ah, and at the other end erect the perpendicular bd equal to the side radius cg. Connect the ends of these perpendiculars by the line cd, and at point 2 erect a perpendicular, meeting cd at e. The length e2 will be the desired third radius. With the compasses set to this radius, find a center i from which a curve can be struck which will be just tangent to the side and end curves. From other centers similarly located the remainder of the ellipse is drawn. Lines drawn from i through h, and from g through i determine the meeting points of the different curves. For narrow ellipses the length of the end radius, ah, should be increased as follows: For an ellipse having its breadth equal to one-half of its length, make ah one-eighth longer. For an ellipse having its breadth one-third of its length, make ah onefourth longer. For an ellipse having its breadth equal one-quarter of its length, make ah one-half longer. For intermediate breadths lengthen ah proportionately. With this modification of the length of the end radius, this method gives curves which blend well together so as to satisfy the eye, and gives a figure which conforms quite closely to the actual outlines of an ellipse. |