(3) If the directrix, focus and eccentricity be given, we may employ the general method for constructing a conic. Let S (fig. 2) be the focus, KX the directrix, X being the foot of the perpendicular from S to the directrix. Divide SX internally at A and externally at A', so that the ratios SA/AX and SA'/A'X are each equal to the eccentricity. Then A, A' are the vertices of the curve. Take any point R on the directrix, and draw the lines RAM, RSN; draw SL so that the angle LSN = angle NSA'. Let P be the intersection of the line SL with the line RAM, then it can be readily shown that P is a point on the ellipse. For, draw through P a line parallel to AA', intersecting the directrix in Q and the line RSN in T. Then since XS and QT are parallel and are intersected by the lines RK, RM, RN, we have SA/AX = TP/PQ = SP/PQ, since the angle PST = angle PTS. By varying the position of R other points can be found, and, since the curve is symmetrical about both the major and minor axes, it is obvious that any point may be reflected in both the axes, thus giving 3 additional points. [Illustration: FIG. 2.] (4) If the axes be given, the curve can be approximately constructed by circular arcs in the following manner:--Let AA', BB' be the axes; determine D the intersection of lines through B and A parallel to the major and minor axes respectively. Bisect AD at E and join EB. Then the intersection of EB and DB' determines a point P on the (true) curve. Bisect the chord PB at G, and draw through G a line perpendicular to PB, intersecting BB' in O. An arc with centre O and radius OB forms part of a curve. Let this arc on the reverse side to P intersect a line through O parallel to the major axis in a point H. Then HA¹ will cut the circular arc in J. Let JO intersect the major axis in O1. Then with centre O1 and radius OJ = OA¹, describe an arc. By reflecting the two arcs thus described over the centre the ellipse is approximately described. ELLIPSOID, a quadric surface whose sections are ellipses. Analytically, it has for its equation x²/a² + y²/b² + z²/c² = 1, a, b, c being its axes; the name is also given to the solid contained by this surface (see GEOMETRY: _Analytical_). The solids and surfaces of revolution of the ellipse are sometimes termed ellipsoids, but it is advisable to use the name spheroid (q.v.).