[alogo] Oscillating Circles of Ellipse

Given the ellipse (e) with axes a, b (a >b) and equation x/a + y/b =1. The circle (c) with center at the origin and radius d = sqrt(a^2+b^2) = sqrt(2*b^2+c^2) is called the [Director] circle of the ellipse (see Director.html ). It is the locus of points viewing the ellipse under a right angle. Jacob Steiner (Werke Bd. II, p. 341) gives an interesting application of it to the determination of the [oscillating circle] of a point A of the ellipse. The recipe to construct the radius of the oscillating circle is the following one:
1) Find the intersection point E of the normal at A with the polar p of A w.r. to the circle (c).
2) Take the symmetric A* of E with respect to A. This is the required center of curvature at the point A, the radius being equal to R = |A*A|, the curvature of (e) at being k = 1/R.

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[1_0] [1_1] [1_2]


An easy calculation for the curvature point A*(x*,y*) in terms of A(x,y) gives the following equation:

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The figure above shows the circle (f) passing through A and F, which is the inverse of A w.r. to (c), hence orthogonal to it. It shows also the locus of A* [Ellipse Evolute], which is the envelope of the normals of the ellipse.

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