ConicsAndSimilarities

Conics and similarities of circles

File CirclesSimilar.html contains a short discussion on the locus of points which serve as similarity centers F of some similarity S mapping a circle (d) to another circle (d'). Here it is shown that, for such a similarity, line EE' joining a point E on (d) with E'=S(E) on (d') envelopes a conic with one focus at F and tangent (under certain conditions) to the two given circles (d) and (d').

[1] From the elementary properties of the hyperbola (see Hyperbola.html ) follows that the tangent line XT and the normal line XN at a point X of the hyperbola (the ellipse as well) intersect the major axis (x-axis above) at two points {T,N} which are harmonic conjugate to the foci {F,F'} of the hyperbola.
[2] It follows that the circle (e) through {F,F',X} passes also through K, the intersection point of the minor axis (y-axis) with the normal line XN. In fact, the previous property implies that XN, XT are bisectors of the angle at X of triangle FXF' and the result follows from that directly.
[3] As a consequence triangles KOF and XGF are similar. More general, the auxiliary circle (c) and a circle (d), tangent at two points of the conic, are similar under the similarity S defined by the following data. The center of S is F. The rotation angle is OFK and the ratio k=OF/OK.
In fact S, defined by these data, maps (c) to another circle centered at K and with radius KX, since triangles OGF and KXF are similar. Thus this circle must coincide with (d).
[4] Since D is on the auxiliary circle and D'=S(D) is such that DD' is orthogonal to FD if follows that DD' is tangent to the hyperbola (ibid).
[5] Selecting another point X'(x',y') of the hyperbola we construct another circle (d'), which, like (d), is similar to the auxiliary circle though a similarity S' centered at F etc.. The composition of similarities S'*S^-1 = S'' is a similarity having the property that for every pair of corresponding points {E, E'=S''(E)} line EE' is a tangent to the hyperbola.
[6] This property has also an inverse: For every pair of circles {d, d'} and a similarity S mapping the one to the other, which is not a homothety, then for E in d and E'=S(E) in d', line EE' envelopes a conic with one focus F at the similarity center and tangent to both circles. The proof of this can be easily deduced by reversing the arguments of the previous constructions. First define another circle, corresponding to (c) and define a similarity S' mapping (c) to (d). Prove for this the analogous property and extend then to the more general case.
[7] By these properties the hyperbola appears as an envelope of a set of circles which are similar to the auxiliary circle under similarities centered at a focus of the conic and the lines joining corresponding points under these similarities are tangent to the conic. Further, any two circles of this set define a unique similarity with this property.
[8] Returning to the remarks made in CirclesSimilar.html , I remark that the shape of the conic (here an hyperbola) and the tangency to the two circles (d) and (d') depends on the relative position of the two circles and the position of the corresponding similarity center F, contained in the circle (f) with diametral points the two homothety centers of the circles.
[8.1] In the case of two non-intersecting circles (d), (d'), lying outside each other, for any point F on (f) different from the two homothety centers we obtain always an hyperbola. The hyperbola is always tangent to the circles (d) and (d').
[8.2] In the case of two non-intersecting circles (d), (d'), lying the one inside the other, and for any point F on (f), different from the two homothety centers we obtain always an ellipse. The ellipse lies always inside the small circle and (f) is divided into four arcs such that for points F lying on these arcs alternatively, the ellipse is totally inside the small circle or is tangent to it. File ConicsAndSimilarities2.html contains a typical illustration of such a case.
[8.3] In the case of two intersecting circles we obtain both cases of hyperbolas and ellipses, depending on the position of F on circle (f). In this case circle (f) passes through the intersection points {M,N} of circles {d,d'}. For points F on (f) lying outside the intersection of two circular discs the conics are hyperbolas. For points F inside the intersection of the two circular discs the conics are ellipses. File ConicsAndSimilarities3.html contains a typical illustration of such a case.

Conics and similarities of circles

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