Fig. 220. - To Draw an Approximate Ellipse with the Compasses, Using Three Sets of Centers.

Fig. 221. - To Find the True Axes of a Given Ellipse.

Fig. 222. - In a Given, Ellipse, to Find Centers by which an Approximate Figure may be Constructed.

77. To Find the True Axes of a. Given Ellipse. - In Fig. 221, let N P O R be any ellipse, of which it is required to find the two axes. Through the ellipse draw any lines, A B and D E, parallel to each other. Bisect these two lines and draw F G, prolonging it until it meets the sides of the ellipse in the points H and I. Bisect the line H I, obtaining the point C. From C as center, with any convenient radius, describe the are K L M, cutting the sides of the ellipse at the points K and M. Join K and M by a straight line, as shown. Bisect M K by the line N O, perpendicular to it. Through C, which will also be found to be the center of N O. draw P R, perpendicular to N O and parallel to K M. Then N O and P R are the axes of the ellipse.

78. In a Given Ellipse, to Find Centers by which an

Approximate Figure may be Constructed__In Fig. 222, let A F B D be any ellipse, in which it is required to find centers by which an approximate figure may be drawn with the compasses. Draw the axes A B and E D. From the point A draw A F, perpendicular to A B, and make it equal to C E. Join F and E. Divide A F into as many equal parte as it is desired to have sets of centers for the figure. In this instance four. Therefore, A F is divided into four equal parts, as shown by P O and G. Divide A C into the same number of equal parts, as shown by R

S T. From the points of division in A F draw lines to E. From D draw lines passing through the divisions in A C, prolonging them until they intersect the lines drawn from A F to E, as shown by D U, D V and D W. Draw the chords U V, V W and W E, and from the center of each erect a perpendicular, which prolong until they intersect as follows: The line perpendicular to W E intersects the center line E D in the point D. Now draw D W and prolong the perpendicular to V W till it intersects D W in K, and draw K V. Prolong the perpendicular to U V till it cuts K V in L and draw L U, cutting A C in the point S. Then D is the center of the arc E W, K is the center of the arc W V, L is the center of the arc V U and S is the center of the are U N. By these centers it will be seen that one-quarter of the figure (A to E) may be Struck. By measurement, corresponding points may be located in other portions of the figure. If correctly done the points U, V and W will be found to fall upon the ellipse, consequently the arcs drawn between those points from the centers obtained cannot deviate much from the correct ellipse. 79. To Draw the Joint Lines of an Elliptical Arch. - First Method. - In a circular arch the lines representing the joints between the stones forming the arch, or the voussoirs as they are properly called, are drawn radially from the center of the semicircle of the arch. In an elliptical arch this operation is somewhat more difficult as the true ellipse possesses no such single point, but, instead, two foci, as has been explained. Therefore, the following course must be pursued: From any point upon the ellipse at which it is desired to locate a joint, as A, Fig. 223, draw a line to each of the foci, as A B and A C. Bisect the angle BAG (Prob. 12 in this chapter), as shown at D, and extend the line D A outside the ellipse, which will be the joint line required.

Fig 223. - First Method. To Draw the Joint Lines of an Elliptical Arch.

80. To Draw the Joint Lines of an Elliptical Arch. - Second Method. - In Fig. 224, A B is one-half the curve of the arch, A C its center line and C B its springing line. Draw A D parallel to C B, and D B parallel to A C. and draw the diagonals A B and C D. From each of the points 1, 2, 3, etc., representing the joints, drop lines vertically, cutting C D. From their intersections with C D carry them at right angles to A B, cutting the springing line C B, as shown by the small figures 12, 22, 32, etc. From the points in C B draw lines through corresponding points in the arch A B, as 12 1, 22 2, 32 3, etc., and continue them through the face of the arch which will be the joint lines sought.

Fig. 224. - Second Method. To Draw the Joint Lines of an Elliptical Arch.

In the case of an elliptical curve made up of arcs of circles, the joint lines would be drawn radially from the centers of the arcs in which they occur.