Given the ellipse with axes a=HE, b=HI (a>b) and equation x²/a² + y²/b² =1. The points D, E are called *Vertices*. The *auxiliary circle* of the ellipse is the one with diameter DE. For each point A on the ellipse the vertical line GA intersects the auxiliary circle at a point F, defining the *eccentric angle* phi = angle(EHF) of point A. For the coordinates (x,y) of A holds x = a*cos(phi) and y = b*sin(phi). The first is obvious, since HF = HE = a. The second follows from the relation:

On the other side FG² = DG*GE, hence AG²/b² = FG²/a², or AG/FG = b/a. Thus FG=a*sin(phi) implies the AG = y = b*sin(phi).

This property dictates also a way to construct the ellipse with axes a and b. Start with a circle of radius a, select a diameter DE and from every point F of the circle draw the perpendicular FG on DE. On it select A, such that AG/FG = b/a, where 0<b<a a fixed number. Then the locus of points A is an ellipse with axes a and b.

It is not necessary to take the projecting axis to be a diameter of the circle. One can produce the same (congruent) ellipse by projecting F to an arbitrary line. This exercise is discussed in Ellipse_Construction2.html .

The intersection of lines I'J', I''J'' at a point K of the major axis is an immediate consequence of the discussion. An analogous property holds also for the interesection points with lines orthogonal to the *minor* (here the vertical) axis of the ellipse.

### See Also

CommonPolar.html

CommonPolar2.html

Director.html

Ellipse.html

Ellipse_Construction2.html

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