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Circles Associated to Pairs of Conjugate Diameters Paris Pamfilos

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Circles Associated to Pairs of Conjugate Diameters Paris Pamfilos INTERNATIONAL JOURNAL OF GEOMETRY Vol. 7 (2018), No. 1, 21 - 36 CIRCLES ASSOCIATED TO PAIRS OF CONJUGATE DIAMETERS PARIS PAMFILOS Abstract. A pair of conjugate diameters of a non-circular ellipse or a hyperbola defines a pair of circles, the points of which are naturally related to the points of the conic of reference by a couple of projective transformations. We prove the existence of these circles and corresponding transformations, as well as some of their fundamental properties. 1Thetwocircles Consider an ellipse κ and a fixed pair of conjugate diameters of it {OS, OS}.Letthe varying tangent tP at P intersect the diameters at the points {T,T }. Joining these with the focal points, we obtain the line intersections {U =(FT,FT ),U =(FT,FT )}.The T τ C S P U F' tP F O U' S' τ' T' κ C' Figure 1: Circles generated by a pair of conjugate diameters discussion in this article is about the geometric loci of these two points, as P varies on the ellipse (conic), proven to be two circles {τ,τ} (See Figure 1). We examine also some of the fundamental geometric properties of these circles. The properties dealt with are about genuine central conics, which are not circles. Below we confine the discussion mainly to the case of hyperbolas, in order to encompass also some special properties related to asymptotes and the conjugate hyperbola of κ. Keywords and phrases: Conic, Hyperbola, Ellipse, Conjugate Diameters (2010)Mathematics Subject Classification: 51N15, 51N20, 53A04, 97G40 Received: 9.11.2017. In revised form: 18.01.2018. Accepted 05.01.2018 Circles associated to pairs of conjugate diameters 22 The general properties though, not involving the asymptotes, and their proofs for non- circular ellipses, are essentially the same, some fundamental differences being discussed in section-6. A key role in the investigation plays the map f of the plane, initially defined only for the points of the conic of reference κ by the rule f(P )=U, mapping κ onto τ. The analogous map f , defined in the same way by the rule f (P )=U , mapping κ onto τ , is essentially the same with f and τ is the symmetric of τ w.r. to the center O of the conic. In fact, using the canonical system of coordinates of the central conic, it is easily seen that f (P )=−f(−P ).Themapf can be extended to a projectivity of the whole plane onto itself and its use facilitates the detection and formulation of various geometric properties of the circles {τ}. Regarding the set of all these circles {τ},wenoticethatitis in a one to one correspondence with the points of the conic of reference κ. In fact, walking in a fixed orientation on the conic, we see that each point S(x1,y1) ∈ κ defines a conjugate direction and an associated circle τ = τS depending on S. Thus, {τS,S ∈ κ} is a one-parameter family of circles, the properties of which appear in all parts of this article. The under discussion configuration is a bit reminiscent of (Ι) ε (ΙΙ) T ε P β P U α U2 Q U F' F' 1 Τ' T F F O β α U Ο Τ Figure 2: Maclaurin generation of a conic ... trying to generalize the Maclaurin-Braikenridge generation of conics ([1, p. 69]), according to which, a vari- able line ε, revolving about the fixed point P , intersects two fixed lines {α, β} at points {T ,T}. These points, connected to two other fixed points {F, F }, define the intersections {U =(FT,FT )} describing a conic (See Figure 2-I). The attempt to generalize, by re- placing lines {ε} with the tangents of a conic, leads to curves of higher degrees (See Figure 2-II). In the case the intersection point Q =(α, β) coincides with the center O of the conic, we can see that points {U} do indeed describe a conic. Our case is the particular one of this, for which, not only Q coincides with the center O of the conic, but also the lines {α, β} represent two conjugate diameters. 2 The projective transformation We start with the rendition of the analogous to figure 1, this time for the hyperbola. Since the property under consideration is invariant under similarities, we can simplify the calculations by assuming that the axes {a, b} of the hyperbola satisfy the equation c2 = a2 + b2 =1, i.e. the distance of the focal points is |FF| =2c =2. Theorem 2.1. Given two conjugate directions {α, β} of a hyperbola κ with focal points {F, F }, consider the intersections {T =(α, tP ),T =(β,tP )} with the tangent tP at P . Then, the intersection of the lines U =(FT,FT )(respectively U =(FT,FT )), defines a correspondence f : P −→ U, (respectively f : P −→ U ), which is the restriction on the hyperbola of a projective map of the plane onto itself. Circles associated to pairs of conjugate diameters 23 U P β α T S U' F' O F tP T' κ Figure 3: The map f : P −→ U Proof. Using canonical coordinates, in which the hyperbola has an equation of the form x2 y2 − =1, assume P =(x, y),α(λ)=λ(x1,y1),β(µ)=µ(x2,y2), (1) a2 b2 where S =(x1,y1) ∈ κ i.e. satisfies the equation for κ,andλ, µ are real parameters of the corresponding lines. The tangent at P has the direction of the conjugate to line OP. Similarly (x2,y2) defines the conjugate direction of (x1,y1) and we can set ([3, p. 167]) 2 2 2 2 tP (u)=P + u(a y,b x), (x2,y2)=(a y1,b x1). (2) A short calculation, using this and equation-(1), shows that the parameter values {u, u} in (2), for which we obtain correspondingly the points {T,T} are − 2 − 2 1 = y1x x1y = b x1x a y1y ⇒ · = u 2 2 ,u 2 2 u u 2 2 . (3) b x1x − a y1y a b (y1x − x1y) a b Using this and performing a somewhat lengthier calculation, gives point U(u1,u2) through a parameter ν along the line F T : 2 ( − ) = + · ( ) = y1 y1x x1y U F ν F T , with ν 2 2 2 . (4) b x +2y1(y1x − x1y) − a b Finally, from equations-(3, 4) and simplifications, deriving from the fact that (x1,y1) satis- fies equation-(1), result the coordinates of U,expressedintermsofthefixedS =(x1,y1), which defines the two conjugate diameters, and the variable on the hyperbola point P (x, y): 2 2 + 2 2 − 2 2 2 = b x1 a y1 b x = b x1y1 u1 2 2 2 ,u2 2 2 2 . (5) b x +2y1(y1x − x1y) − a b b x +2y1(y1x − x1y) − a b The proof follows from the fact that the common denominator and the two numerators are linear functions of the variables {x, y}. Thus, the functions, expressing U in dependence of P (x, y), can be extended to the whole plane using the above formulas, defining then a projective transformation f of the plane onto itself ([5, I, p. 190]). Similar arguments lead also to the proof for the other function f . In this case the corresponding formulas are 2 ( − ) = + · ( ) = y1 y1x x1y ⇒ U F ν FT , with ν 2 2 2 (6) b x +2y1(y1x − x1y)+a b 2 2 + 2 2 + 2 2 2 = b x1 a y1 b x = b x1y1 u1 2 2 2 ,u2 2 2 2 . (7) b x +2y1(y1x − x1y)+a b b x +2y1(y1x − x1y)+a b Circles associated to pairs of conjugate diameters 24 In accordance with the remarks in the introductory section, we should notice that, essentially, there is only one function f depending on S(x1,y1) and, to be more precise and ( ) stress the dependence on S, we should append an index and write f(x1,y1) x, y , instead of ( ) ( )=− (− − ) f x, y .Thenf would be seen to be f x, y f(−x1,−y1) x, y . Notice also that in homogeneous cartesian coordinates (x,y,z), connected to (x, y) by the relations {x = x/z,y = y/z} ([3, p. 65]), the equations defining the maps {f,f−1} take the form of non-zero multiples of the linear transformations by the matrices: ⎡ ⎤ ⎡ ⎤ 2 2 2 2 2 −b 0 b x1 + a y1 x ⎢ 2 ⎥ ⎢ ⎥ f(x ,y ,z )=⎣ 002b x1y1 ⎦ ⎣ y ⎦ , 2 2 2 2 2 b x1 +2y1 −2y1x1 −a b z ⎡ ⎤ ⎡ ⎤ 2 2 2 2 2 2 2 2a b x1y1 −a (a y1 + b x1)0 x −1 ⎢ 2 2 2 2 2 2 2 2 4 ⎥ ⎢ ⎥ f (x ,y ,z )=⎣ b (a y1 + b x1) −2a b x1y1 a b ⎦ ⎣ y ⎦ . 0 −a2b2 0 z Deducing these matrices from equations-(5) we must take into account that (x1,y1) satisfies 2 2 2 2 2 2 2 the equation of the hyperbola, which implies the relation a (2y1 + b )=a y1 + b x1. 3 The circle property Since {τ,τ} are images of the conic κ under the projective transformations {f,f},they are certainly conics. The circle property results by eliminating {x, y} from equations-(5) (resp. equations-(7)). Theorem 3.1. The conics {τ,τ}, defined as images {f(κ),f(κ)} through the projectivities of theorem-2.1, are circles. Proof. Solving the equations in (5) with respect to {x, y},weobtain 2 2 2 2 2 x =(−(2x1y1)u1 +(2x1 − a )u2)/u2,y=((b +2y1)u1 − 2x1y1u2 + b )/u2.
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