. Plane and solid analytic geometry . Fig. 14 m,as THE ELLIPSE 119 A of each successive ellipse are farther away from O, andtheir distances from 0, namely, m 00 = 1- OA = m1^ i \ increase without limit. Thus, as e approaches 1, the ellipseapproaches as its limit the parabola whose directrix is D andwhose focus is F. 10. New Geometrical Construction for the Ellipse. Para-metric Representation. Let it be required to draw an ellipsewhen its axes, AA and BB, are circles of radii a = OA andb = OB, with the origin 0 as thecommon center. Draw any ray from0, making an angle <^ with t
. Plane and solid analytic geometry . Fig. 14 m,as THE ELLIPSE 119 A of each successive ellipse are farther away from O, andtheir distances from 0, namely, m 00 = 1- OA = m1^ i \ increase without limit. Thus, as e approaches 1, the ellipseapproaches as its limit the parabola whose directrix is D andwhose focus is F. 10. New Geometrical Construction for the Ellipse. Para-metric Representation. Let it be required to draw an ellipsewhen its axes, AA and BB, are circles of radii a = OA andb = OB, with the origin 0 as thecommon center. Draw any ray from0, making an angle <^ with the posi-tive axis of X, as shown in the Zfigure. Through the points Q and Bdraw the parallels indicated. Theirpoint of intersection, P, will lie onthe ellipse. For, if the coordinates ofPbe denoted by (x, y), it is clear that \ (1) x = a cos(f), y =h sin<^. From these equations can be eliminated by means of thetrigonometric identity sin2 (ji -f cos2 (^ = 1. y. Fig. 15 Hence (2) ^ + ^ = 62 Conversely, any point {x, y) on the ellipse (2) has corre-sponding to it an angle <^, for which equations (1) are true. Equations (1) afford what is known as a parametric repre-sentation of the coordinates of a variable point {x, y) of theellipse in terms of the jyarameter , y = a sin <^. 120 ANALYTIC GEOMETRY These parametric representations, though little used in Ana-lytic Geometry, are an important aid in the Calculus. The larger of the two circles in Fig. 15 is commonly called the auxiliary ^:{z,ij) circle of the ellipse, and the points R and P are known as corresponding points. The angle ^ is called the eccentric angle.
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