ELLIPSIS, or ELLIPSE, in geometry, a conic section formed by coffing a cone entirely through the curved surface, neither parallel to the base, nor making a subcon trary section ; so that the ellipsis, like the circle, is a curve that returns into itself, and completely encloses a space. See the definitions under the word CONE.
One of the principal and most useful properties of the ellipsis is, that the rectangle under the two segments of a diameter is as the square of the ordinate. In the circle, the same ratio obtains. but the rectangle under the two segments of the diameter becomes equal to the square of the ordinate : Problem 1.—The two axes of an ellipsis being given to describe the curve.
.11Iethod 1.—Figure 1. Let A c be the greater axis, 13G the lesser, cutting each other in the centre it ; and if with the radius A II, or a c, from the point B, an arc E F be described, it will cut A C, at E and F, the feet ; fix two pins at E and F; take a thread equal in length to A C, and fix one end of it to E, and the other to F ; then keeping a pencil at the point D, move such point forward in the same direction, so that the parts D E and E r may continue to lie stretched during the motion, until the describent D come to the point whence it began to move.
Method 11.—Figure 2. Find the foci E and F, as before ; between E and F, take any point, I ; with the radii A 1, 1 c, and the centres E and F, describe arcs cutting each other at a, as also at ii ; then a and it are points in the curve ; in the same inanner, with the sante from the centres F and E, find the intersections i and x. In like manner, if any other point, 2, be taken between E and F ; four other points, L, N, 0, will be obtained, and thus as many more as will be requisite for drawing the curve by hand.
For by the construction of the ellipsis, E D D F, (Figure 1) is equal to K c CF,=EF-F2FC; FA+ AE=EF+2AE;thereforeEF+2PC=EF+ 2 A E: consequently A E is equal to c; hence ED+Dr =- E F 2A E = E F +2FC: that is, the sum of the two lines drawn from the foci, to any point in the curve, is equal to the transverse axis.
Method 111.—Figure 3. To describe an ellipsis with the ellipsograph, or trammel, as it is called by worAmes, the axes A n and C D being given in position, bisecting each other in the centre E.—ln any piece or material, contained between any two parallel planes, cut two grooves, at right angles to each other, in one of the planes ; then provide a rod with three pins, or points, so that at least two may be moveable. and in a straight line with the third : let n F o be the rod, with the points F and o moveable ; making it a equal to the greater semi-axis ; then placing the grooves over the axes, and putting the points F and a in the two grooves, move the point n round, keeping the point F upon the greater axis, A n, and the point a upon the lesser axis c D, until the describent H come to the point where the motion commenced ; and the figure so described, will be an ellipsis. The trammel, used by artificers, consists of two rulers. with a groove in each, so fixed that both grooves may be in the same plane, and at right angles to each other, and that the opposite sides of the cross may be in a plane parallel to those of the grooves. The rod above, is a bar with two moveable cursors, the fixed end is made to hold a pencil, and each of the other two an iron point. made to fill the groove, but capable of sliding freely.
Method IV .—Figure 4. Given one of the axis, A B, and an ordinate, c a, to describe the ellipsis.—Biseet A B at i for the centre ; through i draw E F, parallel to c D; with the distance A, or i 11, from the point n, describe an arc, cutting i v at a ; draw the straight line n a it, cutting i A at u ; then if u D a be conceived to be an inflexible line or rod, the points Ii, a, D, remaining at the same distance in respect to each other ; and if the point it he moved in the axis A B, and the point a in the axis E F, while the describent D, is carried round the centre, 1, Until A conic to the point whence it began to move ; a curve DuE A F, will be described, which will be an ellipsis.