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. (8)

The quadratic equation for (u1,u2) results by replacing these expressions into the hyperbola 2 2 2 2 equation-(1) and doing some simplifications, using the conditions x1/a − y1/b − 1=0 and c2 = a2 + b2 =1. The two equations show that {τ,τ} are two equal circles lying symmetrically w.r to the center O :

2 2 2 2 2 2 2 2 2 2 2 a b (u1 + u2) ± 2(b x1 + a y1)u1 ∓ 4a (x1y1)u2 + a b =0. (9)

From these equations follows that the centers {C, C} of the circles and their radius are Ç å 2 2 2 2 2 2 b x + a y 2x1y1 4x y ± − 1 1 , ,r2 = 1 1 . (10) a2b2 b2 a2b4

Corrolary 3.1. The circles {τ,τ} are orthogonal to the circle σ on diameter FF.

Proof. In fact, the intersections of the circles {τ,τ} with the x−axis are the roots of the quadratic equation w.r. to the variable u1, resulting from equation-(9) for u2 =0.These are equal to −1 2 (bx1 ± ay1) , (11) a2b2 and their product is equal to 1, which is the (square) radius of σ (See Figure 4). Circles associated to pairs of conjugate diameters 25

τ B' ηκ

B C

S

G'F' O F G τ' C'

σ

σ'

Figure 4: The hyperbola-locus of the centers {C, C}

Theorem 3.2. The centers {C, C} of the circles {τ,τ} are on a hyperbola η with the same axes as the hyperbola of reference κ. The hyperbola η passes through the focal points {F, F } of κ and its own focal points {G, G} lie on the circle σ, which is the inverse w.r. to σ of the minor circle of κ i.e. the circle at the origin with radius b = |OB|. Proof. All this results from the equation satisfied by η, which can be deduced from equation-(10) and is b2 η : x2 − y2 =1. (12) a2 Corrolary 3.2. The hyperbola η of the centers {C} of circles {τ} is similar to the conjugate of the hyperbola of reference κ.

Proof. This is immediately» seen from the eccentricity of η, which, under the assumption a2 + b2 =1,is 1+a2/b2 =1/b, whereas the eccentricity of κ is 1/a.Thus,thetwo eccentricities {e, e} satisfy 1/e2 +1/e2 =1, which is characteristic of the eccentricities of a hyperbola and ist conjugate. Notice that the homothety {x = bx, y = by} transforms η to the isometric to the conjugate of κ,hyperbolaκ : x2/b2 − y2/a2 =1. Figure-5 ilustrates the behavior of the circles {τS} for some points S ∈ κ lying in two quadrants of the coordinate system. The figure, easily deducible from formulas (9), shows that all circles corresponding to points S in a certain quadrant contain the focus not lying in this quadrant, are pairwise non-intersecting and, as S tends to infinity in this quadrant, the corresponding circle τS tends to the asymptote contained in this quadrant. In particular, there is no real envelope of this family of circles. As will be seen below, this is different in the case κ is an ellipse, in which such an envelope exists (section-6).

4 Alternative views

Some alternative views of the configuration result from a known property of conjugate diameters ([4, p. 127]), which, for the sake of completeness of the discussion, I formulate and prove in short in the form of the following theorem.

Theorem 4.1. The tangent tP at P of the hyperbola intersects two conjugate diameters {α, β} at the points {T,T} and point U is the intersection of lines U =(FT,FT ). Then triangles {PTF, TTU, P F T } are similar. Analogously {U TT,PFT,PTF} are similar triangles, where U =(FT,FT ). Circles associated to pairs of conjugate diameters 26

S2

S

F' O F

S1

κ

Figure 5: The one-parameter family of circles {τS}

Proof. Since F◊PT = TPF’ , to prove that triangles {F PT,TPF} are similar, it suffices PF
PT to show that PT
= PF . Using the relations of the previous section, this obtains the equivalent form (See Figure 6)

P U β α

T S U'

F' O F

T'

Figure 6: Similar triangles

2 2 ( + )2 + 2 2( 4 2 + 4 2) PF = PT ⇔ c x0 y0 = u a y0 b x0 ⇔ 2 2 2 4 2 4 2 2 2 PT PF u (a y0 + b x0) (c − x0) + y0 4 2 4 2 2 2 2 2 2 2 2 4 2 4 2 2 (a y0 + b x0) ((c + x0) + y0)((c − x0) + y0)=u u (a y0 + b x0) = . a4b4 Replacing in the left side c2 = a2 + b2 and doing some calculation, we see that this is equal to the right side. This shows the similarity of triangles {PTF, PFT }, implying that {P, F, T ,U} are concyclic. This implies, in turn, the similarity of triangles {UTT, PTF}. Analogous is the proof for the other triple of triangles. In the following short sequence of corrolaries we continue with the notation and the conventions adopted sofar. Corrolary 4.1. Holding point P of the hyperbola κ fixed and varying only the point S, defining the two conjugate diameters {α, β} ,thepoints{U, U } describe a circle ξ(K) passing through the focal points {F, F } and having for center the intersection point K of the normal nP of the hyperbola at P with the y−axis. Circles associated to pairs of conjugate diameters 27

β

K U γ

tP α P τ ξ T U' S O N F' F

K' κ T'

Figure 7: Circe ξ described by {U, U } for fixed P and variable S

Proof. The angle FPF÷ being the double in measure to FUF÷ and FU◊F , implies that points {U, U } are on a circle ξ(K), the points of which view the segment FF under a constant angle. The center K of ξ is the point on the medial line of FF viewing this ÷ segment under the same angle FPF . Since the tangent tP is the bisector of the angle FPF÷ it passes through the middle K of the arc FK˚F and KK is a diameter of the circle γ =(KKP ) (See Figure 7).

γ t δ K p U P β α nP ξ Q

T S U'

F' O F N

K' T'

Figure 8: Alternative definition of points {U, U }

Corrolary 4.2. Line UU passes through the intersection point N of the normal nP with the (transverse axis of the hyperbola) x−axis (See Figure 8).

Proof. Using the expressions for {u1,u2,u1,u2} in equations-(5,7), and the fact that (under the assumption c2 = a2 + b2 =1) the normal at P (x, y) intersects the x−axis at N(x/a2, 0), we can do some computing and see that the collinearity condition, expressed through the determinant, is satisfied: u1 u2 1 1 =0 u1 u2 . x/a2 01 Circles associated to pairs of conjugate diameters 28

Corrolary 4.3. Points {U, U } are constructible from the data {F, F ,a,b,α,ξ},bycon- sidering all circles through {F, F } without reference to the hyperbola points.

Proof. Consider a circle γ through {F, F } with diameter KK on the medial line of FF, points {K, N} being the intersections of the normal nP with the y−,resp. x−axis. It is well known that PN/PK = b2/a2 ([2, p. 95]). Hence, given a circle ξ through {F, F }, equivalently circle γ through its center K,pointP can be located on γ from this fixed ratio. Then T is found as the intersection of KP with the line α.ThenU is found through as second intersection of circle ξ with line F T amd point U is found as second intersection of ξ with line U N.

÷ Corrolary 4.4. The tangent tP bisects angle UPU and points {U, U ,Q,N} make a har- monic division. Here Q denotes the intersection point Q =(UU ,tP ).

Proof. By the previous theorem the four points are collinear and by theorem-4.3, the quadrilaterals {TPUF, TPUF } are cyclic. Thus {TPU÷, TPU÷} are respectively cople- mentary to {TFU,TFU},whichareequal.

Corrolary 4.5. Line δ = OQ is independent of the position of P , depending only on S and coinciding with the harmonic conjugate of the x−axis w.r. to the conjugate diameters of reference {α, β}.

Proof. The coordinates of Q =(UN,tp) can be readily seen to be

2 2 + 2 2 2 2 = a y1 b x1 = b x1y1 Q1 2 2 2 2 ,Q2 2 2 2 2 . (a y1 + b x1)x − 2x1y1y (a y1 + b x1)x − 2x1y1y

Then, for the slopes {m1,m2,m3} respectively of lines {α, δ, β} follows 2 2 2 − = y1 = b x1y1 = b x1 ⇒ m1 : m1 m2 = −1 m1 ,m2 2 2 2 2 ,m3 2 . x1 a y1 + b x1 a y1 m3 m3 − m2

B β t τ S

α C S A F' B' O F S' M C' τ' σ

A'

Figure 9: Lines {FS,FS} tangent respectively to {τ,τ}

Corrolary 4.6. The lines {FS,FS} are respectively tangent to the circles {τ,τ}.IfS is the symmetric of S w.r. to the origin O,thenalso{FS,FS} are respectively tangent to the circles {τ,τ}. Circles associated to pairs of conjugate diameters 29

Proof. Let us see what happens to U ,whenP coincides with S (See Figure 9). Then, from the proper definition of points {T,T}, follows that that T coincides with S on line α and T coincides with the point at infinity of β, which is conjugate to α, hence parallel to the tangent of the hyperbola at S.If{A, B} denotes the corresponding positions of {U ,U} for P = S, then lines {AF, BF } pass through T , hence they are parallel to β.By theorem-4.1 the angles {F÷AF , F÷BF} are equal and the triangle SBF is isosceles. The symmetry of the configuration implies that triangle AAF is also isosceles and the medial line of AA passes through F and C. On the other side, the second point of intersection of line AF with the circle τ is the symmetric of A relative to the medial line of AA (see figure-13 on the symmetry of the configuration), hence coincides with A and AF is tangnet to τ . An alternative proof could be given by applying formulas-(5) and showing that the lines {FS,FS} are the images under f of the tangents of the hyperbola at points {S, S}.

τ

τ1

η C1 F' F

σ ζ χ σ1

Figure 10: Construction of the circle τ

From the corollary follows a convenient way to construct the circles {τ,τ}, correspond- ing to the conjugate diameters defined by OS. For this, it suffices to use their orthogonality to the circle σ and their tangency to the lines {ζ = FS,η = FS} for τ and {F S, F S} for τ . In fact, consider the inversion g w.r. to the circle χ(F ) with radius FF (See Figure 10). This transforms circle σ to the line σ1 orthogonal to FF at F and the wanted circle τ to the constructible circle τ1. Latter is tangent to the lines {η, ζ} and has its center at the intersection C1 of σ1 with the bisector of the angle of these two lines. The wanted circle is τ = g(τ1).

κ σ P U τ T' S ε Β F Α O A' F' B'

T

Figure 11: Hyperbolas from a circle and a point

From the corollary follows also an inverse view of the whole configuration, in which we construct an hyperbola κ from a given circle, forcing this circle to play the role of τ.In fact, given a circle τ and a point O outside it (See Figure 11), consider first the circle σ Circles associated to pairs of conjugate diameters 30 orthogonal to the given one and centerred at O. Consider then an arbitrary line ε through O intersecting the circle at two points {A, B}. Take then the intersection F of the line with σ,lyingbetween{A, B} and the symmetrics {A,B,F} w.r. to O of the circle and all these points. Selecting an appropriate intersection point S of the tangents to the two circles from {F ,F} and using corollary-4.6, we can easily see that the hyperbola κ,with focal points {F, F }, passing through S, has the given circle τ playing the role of τS. Theorem 4.2. The radical axis of the circles {τ,τ } intersects the tangent tA at the vertex A of the hyperbola at a point R, such that the lines {RC, RC} are tangent to the circle σ. Proof. The radical axis of {τ,τ} is the medial line of CC (See Figure 12), which using

R P

τ σ β J α U C U' T S F F' O A C'

T' τ' V

Figure 12: The tangents at the intersections of circles {τ,σ} equation-(10) leads to the determination of R : Ç å a2y2 + b2x2 R = a, a 1 1 . 2x1y1 Then, under our permanently holding assumption c2 = a2 +b2 =1,wehavetotestthatthe distance of the origin O from lines {CR,CR} is 1, which is a straightforward computation. This property allows the determination of pairs of circles (τ,τ) by considering the tangents to σ from arbitrary points R of the tangent tA at the vertex A of the hyperbola. The centers {C, C} of the circles are the intersections of these tangents with the orthogonal to line OR at O and {τ,τ} are the orthogonal circles to σ centerred at these two points. Corrolary 4.7. The lines {CU,CU } intersect at a point V on the parallel to the y−axis through P (See Figure 12). Proof. For this it suffices to calculate the x−coordinate of V and see that, under the assumption a2 + b2 =1, it coincides with the corresponding coordinate of P . Figure-13 ilustrates the symmetry of the configuration, repeating the previously de- scribed constructions a second time for the symmetrics {S,P} of {S, P } relative to the center of the hyperbola. Regarding the notation, points {U1,R1,...} are the symmetrics of {U,R,...} and {U2,R2,...} are the symmetrics of {U ,T ,...} relative to the center O of the hyperbola. There are various consequences of this symmetry, like for example the parallelity of pairs of lines {(UU2,U U1), (TT2,T T1), (RR ,R1R2), ...},forwhich,inmost cases, the proof is trivial and is left as exercise. For example, the parallelity of (RR ,R1R2) is trivial, but the proof of their parallelity to tP is not so obvious and will be handled below (theorem-5.3). The figure ilustrates also some general theorems on conics left as exercises, like, for example, the fact that the lines {TT2,T T1} are tangent to the hyperbola. Circles associated to pairs of conjugate diameters 31

β U T P σ 2 τ α R 2 U' C R T S F F' O R1 τ' S' T1 C'

U2 R'

U P' T' 1

Figure 13: The symmetry of the configuration

5 Using the projective transformation

Here we discuss the geometric properties of the projective transformation f,definedin section-2. The first part deals with the (so called, singular line of f) line ε,sendto infinity by f. All lines, passing through a definite point X of this line, map via f to lines parallel to the direction determined by f(X). The line ε is expressed through the common denominator of the rational functions defining f through the equations-(5), by setting it equal to 0: 2 2 2 b x +2y1(y1x − x1y) − a b =0. (13) The properties of this line, stated in the next theorem, derive from relatively simple cal- culations on the basis of equations (10) and (13) and I omit their proof.

τ β σ C ε δ

E α W G F' Η S F O

D ζ' ζ C' ξ' ξ H'

Figure 14: The line ε sent to infinity by f

Theorem 5.1. With the notation and conventions adopted sofar, the following are valid properties (See Figure 14). 1. The line ε is orthogonal to the line of centers CC of the circles {τ,τ}. Circles associated to pairs of conjugate diameters 32

2. The line ε is the polar w.r. to the hyperbola of the point E, which maps via f to the center C of τ. 3. Point E is the symmetric w.r. to the origin O of D on the parallel to the (conjugate) y−axis, such that the intersection point G with the (transverse) x−axis divides the segment in the ratio CG/GD = a2/b2, the coordinates of D being

1 2 2 2 2 2 D = − · ( b x1 + a y1 , 2b x1y1 ). (14) a2b2

4. Line ED maps via f to line CD and coincides with the harmonic conjugate δ of the x−axis w.r. to the two lines {α, β} definining the conjugate directions of reference (we met already δ in corollary-4.5). 5. More generally, every line ζ through the intersection point W =(ε, ED) maps via f toalineorthogonaltothex−axis. 6. In particular, the parallels to {α, β} through W map via f to the parallels to the y−axis through the focal points {F, F } and the orthogonal to the x−axis through W maps to the x−axis. 7. Line ε passes through the points {H =(α, ξ),H =(β,ξ)}, which are the intersec- tions of the directrices {ξ,ξ} of the hyperbola with the conjugate diameters of refer- ence {α, β}. 8. The x−axis is the singular line, send to infinity, by the inverse projectivity f −1, equivalently, the x−axis is the image by f of the line at infinity.

Q θ ξ' β ξ κ α

H S M

F'A'B' B A F ε

H'

Figure 15: The hyperbola θ enveloping lines {ε : S ∈ κ}

By the way, notice that all these lines, {ε = HH },forvaryingS(x1,y1) on the hyper- bola κ, are tangents to a hyperbola θ, homothetic to κ, hence with the same eccentricity, asymptotes and conjugate directions with those of κ.Thehyperbolaθ has its focal points at the vertices {A, A} of κ and its own vertices at the intersections {B,B} of the x−axis with the directrices {ξ,ξ} of κ (See Figure 15).

Corrolary 5.1. The images under f of the asymptotes {µ, ν} are the tangents {ν,µ} to τ from point D, which are respectively parallels to {ν, µ}, their contact points {I,I} coinciding with the intersection points of the circle τ with the x−axis. Also every line passing through the intersection point {V,V } of ε with an asymptote maps to a parallel to the other asymptote.

Proof. Since the asymptotes pass through O, their images pass through f(O)=D (See Figure 16). The asymptotes are the tangents at infinity of the hyperbola and f maps the line at infinity onto the x−axis (theorem-5.1/6 ). Hence the images under f of the points Circles associated to pairs of conjugate diameters 33

ν' ν β μ' ε μ

τ V E C α

ΙΙF' ' S F O V'

D σ

Figure 16: The images of the asymptotes at infinity defined by the asymptotes are the intersection points {I,I} of the circle τ with the x−axis and the images of the asymptotes coincide with these tangents. In order to show the stated parallelity of {µ,ν} to the asymptotes {µ, ν} we have to examine their directions {DI, DI}, which is an easy calculation on the ground of formulas (11) and (14). The second claim follows from the first, since all lines passing through a fixed point of ε map to parallels and one of this parallels is, by the first claim, parallel to the appropriate asymptote. Next theorem identifies the images f(X) for various points X of the basic configuration involved in the definition of points U,aswellastheimagesf(ζ) for various lines of this configuration. Again the proofs are simple applications of the formulas established in section-3 and I live them as exercises.

U β B δ 2 J B P E α τ C F1 S

F'F2 A' O A F J' B1 σ B' D

tP

Figure 17: The images of some lines under f

Theorem 5.2. With the notation and conventions adopted sofar, the following are valid properties (See Figure 17). 1. The x−axis maps under f to the radical axis JJ of circles {σ, τ }, which passes through point D. 2. The focal points {F, F } maptopoints{F1,F2}, such that lines {FF1,F F2} pass through C. 3. The tangents at the vertices {A, A} of the hyperbola map to the tangents {OJ, OJ} of τ from the point O. Circles associated to pairs of conjugate diameters 34

4. The y−axis maps under f to line δ = DE. 5. The directrices {ξ,ξ} w.r. to {F, F } map under f to the conjugate lines {β,α}. 6. The lines {α, β} map under f to lines {DF ,DF}, which are parallel correspondingly to {β,α}. 7. The intersection points {B,B } of the y−axis with the circle σ maptopoints{B1,B2} on δ, which are harmonic conjugate relative to {O, D} and such that lines {CB,CB} pass respectively through {B2,B1}.

λ P τ β

W σ α U R T S U' O F' F R' D T' λ' W' tP τ'

Figure 18: Line RR parallel to the tangent tP and passes through D

Theorem 5.3. With the notation and conventions adopted sofar, the following are valid properties (See Figure 18). 1. The parallels from P to the conjugate directions {α, β} intersect the hyperbola at points {W, W }, which map under f to the second intersections {R,R} of lines {UF,UF} with τ. 2. The line WW maps under f to line RR, which passes through D and is parallel to the tangent tP of the hyperbola at P . 3. The circles {λ =(TTU),λ =(TTU )} are equal and respectively tangent to {τ,τ} at {U, U }. Proof. Nr-1 derives from f(P )=U and the computation of f −1(F ) through the corre- sponding matrix of section-2. This matrix applied to the projective coordinates [−1, 0, 1] of F , produces a multiple of [x1,y1, 0], thereby proving the claim for the line through P , which is parallel to α. Analogously is proved the claim for the parallel from P to β,this time applying the matrix to [1, 0, 1], which represents F . Nr-2 comprises three claims. The first, about WW mapping onto RR,isanimmediate consequence of nr-1.Since{PW,PW} are parallel to conjugate diameters, it is easily seen that WW passes through the origin O. This implies that line f(WW)=RR passes through D = f(O), as claimed. For the third claim, on the parallelity of RR to tP ,after several attempts with various methods, I realized that the following method minimizes the calculations. First, describe line WW through scalar multiples of the vector a2b2 · W ,creatingthe parameterized line 2 2 2 2 ζ(t)=t · (a b · P − 2(b x0x1 − a y0y1) · S) and its image η(t)=f(ζ(t)). (15) Circles associated to pairs of conjugate diameters 35

( ) Doing some computation weÅ come to the expressionã of the line η t in the form At + B E η(t)= , , with constants Ct + D Ct + D 2 2 2 2 2 2 A =2x1(a y0y1 − b x0x1)+a b x0,B= a − 2x1, 2 2 2 C = a b x0,D= a ,E= −2x1y1, 1 2x1y1 2 2 and η (0) = (AD − BC, −EC)= (a y0,b x0), D2 a2 the last vector being parallel to the tangent tP , as claimed. Nr-3 results from nr-2, as far as the tangency of the circles is concerned. The equality of the circles {λ, λ} is a consequence of the equality of the angles {TUT÷, TU◊T } viewing the segment TT (theorem-4.1).

6 The case of ellipses

Several properties, discussed so far for hyperbolas, transfer, together with their proofs, almost verbatim to the case of ellipses. There are though some differences, which we examine here. Again, because of the invariance of the subject under similarities, and in order to simplify the calculations involved, we restrict ourselves to the case of axes {a, b} of the ellipse satisfying the condition c2 = a2 −b2 =1. Doing the same calculations with those of section-2 we come to the slightly different from equation-(5) corresponding expression for the projective transformation U(u1,u2)=f(P (x, y)) for the case of ellipses: 2 2 − 2 2 + 2 −2 2 = a y1 b x1 b x = b x1y1 u1 2 2 2 ,u2 2 2 2 . (16) −b x +2y1(y1x − x1y)+a b −b x +2y1(y1x − x1y)+a b Using this, the equation of the circles {τ,τ} of section-3 take now the sightly different form 2 2 2 2 2 2 2 2 2 2 2 a b (u1 + u2) ± 2(b x1 − a y1)u1 ± 4a (x1y1)u2 + a b =0. (17) From these equations follows that the centers {C, C} of the circles and their radius are Ç å 2 2 2 2 2 2 a y1 − b x1 −2x1y1 4x1y1 ± , ,r2 = . (18) a2b2 b2 a2b4 Similarly to hyperbolas, also in the case of ellipses the circle σ on diameter FF intersects orthogonally all circles {τ}. This is immediately seen from the previous formulas, from which the power of the origin w.r. to τ is computed and seen to be |C|2 − r2 =1. A difference to the hyperbolas, is that, in this case the x−axis does not intersect the circles, so that the focal points {F, F } are allways outside the circles {τ,τ}.Infact,the intersections of these circles with the x−axis are the roots of the quadratic equation w.r. to the variable u1, resulting from equation-(17) for u2 =0. The discriminant though of 2 2 2 2 this quadratic can be easily seen to be negative and equal to −4a b x1y1. Also the centers of the circles in this case satisfy the equation of an ellipse: b2 (xb)2 (yb)2 x2 + y2 =1. ⇔ + =1, (19) a2 b2 a2 the equation showing that this is similar to the ellipse κ and passes through its focal points {F, F }. Working also with equations-(17) we find that the envelope ([6, p. 75]) of all these circles {τ} is, in this case, the couple of circles (See Figure 19) y x2 + y2 ± 2 − 1=0, with center at (0, ±1/b) and radius r2 = a2/b2. (20) b Circles associated to pairs of conjugate diameters 36

τ

C

κ F' F O

C' τ'

Figure 19: The locus of centers and the envelope of circles {τ}

References

[1] M. Chasles., Traite de Sections Coniques, Gauthier-Villars, Gauthier-Villars, Paris, 1865.

[2] Arthur Cockshott and rev. F.B. Walters., Geometrical Conics, MacMillan and Co.,London, 1891.

[3] George Salmon., A treatise on Conic Sections, Longmans, Green and Co., London, 1917.

[4] Charles Taylor., Geometry of Conics, Deighton Bell and Co, Cambridge, 1881.

[5] Oswald Veblen and John Young., Projective Geometry vol. I, II,GinnandCompany, New York, 1910.

[6] Robert Yates., A handbook on curves and their properties, J.W.Edwards, Ann Arbor, 1959.

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS UNIVERSITY OF CRETE HERAKLION, 70013 GR E-mail address: [email protected]