On Inversions in Central Conics

INTERNATIONAL JOURNAL OF GEOMETRY Vol. 9 (2020), No. 1, 40 - 51

ON INVERSIONS IN CENTRAL CONICS

ROOSEVELT BESSONI AND GUY GREBOT

Abstract. We use plain euclidean geometry to analyse the structure of an inversion in an ellipse or in a hyperbola. By this formulation, results on inversions in circles are directly extended to results on inversions in ellipses.

1. Introduction

Jacob Steiner is quoted in [5] as the first mathematician to formalize the bases of inversive geometry in a text dated from 1824 and published after his death by Bützberger. In 1965, the mathematician Noel Childress [1] extended the concept of circular inversion to central conics, ellipse or hyperbola. In 2014, Ramirez [6][7] showed other properties concerning inversion in ellipses and Neas [4], in 2017, looked at anallagmatic curves under inversion in hyperbolae. Inversion in circles is a geometrical topic which is usually developped without analytic geometry [8]. Strangely enough, we could not find any work concerning inversion in central conics using plain euclidean geometry arguments. In the studied publications the basic framework is analytical geometry. Also, as far as we could reach, the structure of an inversion in a central conic is not mentioned in the cited works and it seems to us that the analytical framework tends to hide this property. In this communication, we use plain euclidean geometry to analyse the structure of an inversion in an ellipse or in a hyperbola. We show that an inversion in a central conic is given by the composition of compressions and an inversion in a circle or in an equilateral hyperbola; in this sense, an inversion in a central conic is just an affine deformation of an inversion in a circle or in an equilateral hyperbola.

————————————– Keywords and phrases: Inversion, compression, inversion in central conics. (2010)Mathematics Subject Classiﬁcation: 51M04, 51M05 Received: 03.09.2019. In revised form: 11.02.2020. Accepted: 12.12.2019 On inversions in central conics 41

As a byproduct, it is shown that to each circle corresponds a family of inversions in ellipses which admit this circle as an auxiliary circle. In the same way, to each equilateral hyperbola corresponds a family of inversions in hyperbolae with same auxiliary circle. In the next section, we deﬁne inversion and compression and we state and prove an important result about compressions in parallel lines. We show the main result of this communication in section 3. Section 4 treats the immediate consequences of the main theorem as well as the proof of a result adapted from a theorem about inversion in circles. We will use the letters a and b to refer to the major and the minor axes, respectively. We shall denote by AB the length of a segment AB and the algebraic measurement (signed length) of such segment will be indicated by AB. The software GeoGebra was used to create the ﬁgures.

2. PRELIMINARY RESULTS In what follows, we will consider Π to be an Euclidean plane. Definition 2.1. Let ̺ be a central conic in Π with center O / ̺. The inversion in the central conic ̺ is the transformation I̺ of Π which∈ associates, to each point P = O in Π, a point P ′ on the ray −−→OP , such that 6 2 (1) OP OP ′ = OQ , where Q ̺ −−→OP . ∈ ∩ It follows directly from this definition that: a point P Π O has an ∈ \ { } image under the inversion in ̺, I̺(P ), only if the ray −−→OP intercepts ̺; I̺ is a 1 1 mapping; if ̺ is an ellipse or a circle, the inversion in ̺, I̺, is defined− for all points in Π O ; if ̺ is a circle of center O, OQ is its radius and the righthand side of\ (1) { is} constant; I̺(P )= P if, and only if, P ̺; if P ′ = I̺(P ) then P = I̺(P ′); a straight line through O that intercepts∈ ̺ is invariant under I̺. For more properties of inversions in circles and inversions in central conics, we refer the reader to the references [8] and to [1], [4], [6] and [7] and references therein. It is to be noted that the treatment of inversions in hyperbolae is more delicate than for inversions in ellipses. This is due to the fact that the definition allows transforming only points that lie in the region limited by the assymptots and which contains the vertices. Although this remark does not affect the results presented below, it contributes to the lack of symmetry between the properties of inversions in hyperbolae and properties of inversions in ellipses. As will be seen in the next section, the main result of this communication expresses that inversions in ellipses and hyperbolae are related to inversions in circles and equilateral hyperbolae, respectively, via a compression. A compression is an affine transformation defined as follows [3]. Definition 2.2. Let c = 0 be a real number and let l be a straight line in 6 Π. The compression τl,c with ratio c over l is the transformation that maps 42 Roosevelt Bessoni and Guy Grebot a point M Π onto the point M ′ according to the vector equality ∈

′ P−−−→ M = c −−→P M, where P l is the orthogonal projection of M over l. ∈ Lemma 2.1. There is a compression that maps an ellipse with center O onto a circle centered in O. E

Proof. The given ellipse is deﬁned by P F1 + P F2 = a, for all P , E ∈ E where F1 and F2 are its foci and F1F2 = c. Let τ be the compression over the focal axis of with coeﬃcient k. Then, for every P , P ′ = τ(P ) is ′ E ∈ E such that P−−−→1P = k −−→P1P , where P1 is the orthogonal projection of P over the focal axis of the ellipse . Figure 1 illustrates these settings. From E

P 0

E P

A F1 O P1 F2 B

Figure 1. Illustration of Lemma 2.1.

2 2 P F1 + P F2 = a , we get 2 2 2 2 2 2 P F1 P F2 = a P F1 + P F2 , h − i 2 2 2 4 2 2 2 2 2 4 P F1 P F2 = a 2 a P F1 + P F2 + P F1 + P F2 , − 4 4 2 2 2 2 2 4 P F1 + P F2 2 P F1 P F2 = 2 a P F1 + P F2 a , − − 2 2 2 2 2 2 4 (2) P F1 P F2 = 2 a P F1 + P F2 a . − − On inversions in central conics 43

Since P P1 F1F2 and the points P1, F1 and F2 are collinear, we have ⊥ 2 2 2 2 2 2 2 2 2 P F1 P F2 = P1F1 P1F2 = P1F1 P1F2 − − − 2 2 = P1F1 P1F2 P1F1 + P1F2 , − P1F1 = P1O + OF1,

P1F2 = P1O + OF2. So equation (2) can be written as 2 2 2 2 c 2 2 4 4 c OP1 = 2a + 2.OP1 + 2.P1P a , 2 − 2 2 2 2 2 2 2 2 (3) 4 c a OP1 = 4a P1P + a c a . − − 2 ′2 ′2 ′2 2 2 As OP1 = OP P1P = OP k P1P , equation (3) becomes − − 2 2 ′2 2 2 a 2 a OP k P1P = P1P + , − (c2 a2) 4 2 − 2 ′2 a 2 2 a OP = P1P k + . − 4 c2 a2 − a a Since a > c, we can put k = and we obtain OP ′ = . This is √a2 c2 2 what we claimed for, i.e. that the compression− over the focal axis of and a E ratio k = takes the point P onto the point P ′ on the circle of √a2 c2 ∈E a − radius centered in O. 2 An ellipse with center O has two auxiliary circles centered in O: one has radius equal to the semi-major axis and the other has radius equal to the semi-minor axis of the ellipse. In the same way as the preceeding lemma exhibits the compression that takes the ellipse onto the outer auxiliary cir- √a2 c2 cle, it can be shown that the compression with ratio k = − and axis a on the perpendicular to the focal axis through O, takes the ellipse onto the inner auxiliary circle.

Lemma 2.2. There is a compression that maps a hyperbola onto an equilateral hyperbola. H

Proof. Let t1 and t2 be the assymptots of and let r be its focal axis. Let H τc be the compression over r with ratio c such that t1 and t2 are mapped onto s1 and s2, respectively, where s1 s2. ⊥ Let A1 and A2 be the vertices of and let O be its center. According H ′ to [2], proposition 31, the hyperbola with vertices A1 and A2 and with H assymptots s1 and s2 is uniquely deﬁned, as illustrated in Figure 2. If p is the parameter of , we have [2] H ′2 LL = p A1A2, ′ ′ where L t1, L t2 and A1 LL . ∈ ∈ ∈ 44 Roosevelt Bessoni and Guy Grebot

s1 H 0

Q0 t1

L 1 L A 2 A 1 V r O

L 0

0 t2 L 1

Figure 2. Illustration of Lemma 2.2.

′ ′ Let L1 = τc(L) and L1 = τc(L ). Then ′ 2 2 ′2 L1L1 = c LL 2 = c p A1A2 which means that the parameter of ′ is given by p′ = c2p. Using the ′ ′ H notation of [2], the ordinate Q at the point V on the line A1A2 is ∈H 2 A V A V Q′V = p′ 1 2 A1A2 A V A V = c2p 1 2 A1A2 2 = c2 QV , where Q is the ordinate relative to V . We then have ∈H Q′V = c QV . Since V , Q and Q′ are colinear, we can state that ′ Q−−→V = c −−→QV , which means that ′ = τc ( ) H H as we whished to prove. On inversions in central conics 45

Theorem 2.1. Let l1 and l2 be two parallel straight lines in Π and let R k1, k2 0 . The compressions τl1,k1 and τl2,k2 satisfy ∈ \ { }

(4) τl1,k1 τl2,k2 = Tu τl1,k1k2 , ◦ ◦ (5) = Tv τl2,k1k2 , ◦ where Tu and Tv are translations along the vectors u = k1(1 k2)−−−→P1P2 and − v = (k1 1)−−−→P1P2, respectively, P1 l1 and P2 is the orthogonal projection − ∈ of P1 over l2.

′ ′′ ′ ′ Proof. Let M = τl2,k2 (M) and M = τl1,k1 (M ). It follows that P−−−→2M = ′′ ′ k2−−−→P2M and P−−−→1M = k1P−−−→1M , where P1 and P2 are the orthogonal projec- ′ tions of M over l2 and of M over l1, respectively. From these, we obtain

′′ ′ P−−−→1M = k1P−−−→1M ′ = k1 P−−−→2M + −−−→P1P2 = k1 k2−−−→P2M + −−−→P1P2 = k1k2 −−−→P1M −−−→P1P2 + k1−−−→P1P2 − which gives ′′ P−−−→1M = k1k2−−−→P1M + k1(1 k2)−−−→P1P2. − We also have

′′ P−−−→1M = k1 k2−−−→P2M + −−−→P1P2 ′′ P−−−→2M = k1 k2−−−→P2M + −−−→P1P2 −−−→P1P2 − and so ′′ P−−−→2M = k1k2−−−→P2M + (k1 1)−−−→P1P2. −

As a consequence of this last theorem, if k1 k2 = 1, we have τl1,k1 =

Tv τl2,k1 , whenever l1 l2. From this remark and the preceeding lemmas, we◦ can state the followingk corollary.

Corollary 2.1. The compression τl,k over l with ratio k maps (1) two equilateral hyperbolae with axis parallel to l onto homothetic hyperbolae; (2) two circles onto homothetic ellipses. 46 Roosevelt Bessoni and Guy Grebot

3. MAIN RESULT The main result of this communication is expressed by the following theorem.

Theorem 3.1. Let be a central conic with major axis a and minor axis b. The inversion withV respect to this conic is expressed by the product

(6) IV = τ 1 I̺ τc, c ◦ ◦ a where τc is the compression over ’s focal axis with ratio c = , and I̺ is b V a either the inversion in the circle concentric to with radius , if is an V 2 V ellipse, or the inversion in the equilateral hyperbola with same focal axis and major axis a as , in case is a hyperbola. V V We shall call ̺, such that IV = τ 1 I̺ τc, the associated curve to . c ◦ ◦ V

′ ′ ′ ′′ Proof. Let P = IV (P ), Q = τc(P ) e P = τc(P ) and ̺ be the image of V by the compression τc. By deﬁnition, we have

2 OP OP ′ = OH where H is the point of intersection between the ray −−→OP and . Let τc(H)= H′ ̺. Figure 3 illustrates these settings. V ∈ Q′

◌̺ H ′ P ′′ P ′ V H O P P0

Figure 3. Illustration of Theorem 3.1 in the case is an ellipse. V On inversions in central conics 47

Since a compression is an aﬃne transformation, it preserves ratios of colinear segments. Thus OP OP ′′ OH OH′ = and = . OP ′ OQ′ OP ′ OQ′ From these equations, we obtain ′2 OP OQ 2 (7) OP ′′ OQ′ = = OH′ . OP ′ Again, using the deﬁnition of inverse in a central conic, equation (7) means that Q′ = I̺(P ′′) and so ′′ τc−1 I̺ τc(P ) = τc−1 I̺(P ) ◦ ◦ ◦ ′ = τc−1 (Q ) ′ = P = IV (P ) which completes the proof that

IV = τ −1 I̺ τc. c ◦ ◦

4. CONSEQUENCES OF THE MAIN THEOREM

From Theorem 3.1 we see that each circle ̺ corresponds to a familly of inversions in concentric ellipses i. Likewise, each equilateral hyperbola ̺ V corresponds to a familly of inversions in concentric hyperbolae i. Two mem- bers of such a familly, which are inversions in concentric ellipses/hyperbolaeV 1 and 2, satisfy V V 1 1 τl1,c1 IV1 τl1, = I̺ = τl2,c2 IV2 τl2, , ◦ ◦ c1 ◦ ◦ c2 where τli,ci , i = 1, 2 is a compression of ratio ci over the axis li. Consequently, we have

1 1 (8) IV1 = τl1, τl2,c2 IV2 τl2, τl1,c1 . c1 ◦ ◦ ◦ c2 ◦ 1 Since τl2, τl1,c1 ( 1)= 2, it is clear that the composition of compressions c2 ◦ V V 1 φ = τl2, τl1,c1 is the composition of a rotation about the center and a c2 ◦ c compression of ratio 1 . c2 In the case of inversions in hyperbolae, the families are disjoint. But, in order to have disjoint families of inversions in ellipses, we must consider the circles to be only circunscribed auxiliary circles or only inscribed auxiliary circles. The properties of inversions in ellipses cited in [1] and [6] are easily seen to be valid, since a compression preserves straight lines and maps circles onto ellipses. For example, the image of a straight line l by the inversion in an ellipse 1 would be the image, under the compression τ 1 , of the image V c of τc(l) by the inversion in the associated circle to 1. As such, it has to be V an homothetic ellipse to 1 in the case where l does not pass through the centre, or a straight line otherwise.V 48 Roosevelt Bessoni and Guy Grebot

Another interesting example, cited in [7], concerns the transformation of a pair of perpendicular straight lines under inversion in an ellipse 1 with center O. Suppose the point of intersection A, between the perpendicularV lines s and t, is distinct from the center O of inversion and that O / s. Then ∈ the images of τc(s) and τc(t) by the inversion in the associated circle to 1 will be either V

(1) one circle with tangent at O parallel to τc(s) which intercepts τc(t); (2) two circles through O with tangents at O parallel to τc(s) and τc(t), depending on whether t passes through O or not, respectively. Upon transformation under τ 1 , the tangents at O are mapped onto perpendicular lines c through O and circles are mapped onto ellipses homothetic to 1. As the result of the inversion we get an ellipse and a straight line or two homotheticV ellipses that intercept perpendicularly at O. Another consequence of Theorem 3.1 concerns anallagmatic curves under inversion in a central conic . V Corollary 4.1. A curve is invariant by the inversion in a central conic C if, and only if, τl,c( ) is invariant by the inversion in ̺, the associated Vcurve to . C V Proof.

IV ( )= τ 1 I̺ τl,c( )= τl,c( )= I̺ τl,c( ). C l, c ◦ ◦ C C ⇔ C ◦ C We end this communication by proving a result about inversion in ellipses which is a new complement to the properties already published in the literature. It is adapted from a theorem cited in [8] that expresses that lines and circles can be mapped into lines or concentric circles by an inversion in a circle. Theorem 4.1. Under an inversion in an ellipse , any two ellipses homothetic to , or a line and an ellipse homothetic to E, can be transformed into concurrentE lines, parallel lines, or two concentricE ellipses. Proof.The proof will be developped by considering three cases, depending on whether the given objects (i.e. two ellipses or one ellipse and a line) have a single point in common, two points in common or no point in common.

Case 1 - a single point in common: Let 1 be an ellipse and let E 2 be either an ellipse homothetic to 1 or a straight line such that E E 1 2 = O . Figure 4 illustrates this case when 2 is a straight line.E ∩E { } E Let r be the straight line through O parallel to 1 major axis. By E lemma 2.1, the compression τr,c over r with ratio c maps 1 onto a ′ ′ E circle 1 and τr,c ( 2)= 2 is either a circle or a straight line such C′ ′ E C that 1 2 = O . Let Γ be a circle with center O and let IΓ be C ∩C { } ′ ′ the inversion in this circle. IΓ maps 1 and 2 onto two parallel ′′ ′′ C C lines 1 and 2, respectively, according to [8]. The compression C C τr,c−1 maps parallel lines onto parallel lines and it maps Γ onto an ellipse homothetic to 1, from corollary 2.1. E E On inversions in central conics 49

′′ ′ ′′ C 2 = C 2 E2 C 1

O r

E1 ′ C 1

Figure 4. Ellipse 1 and tangent line 2. E E

Since we have IE = τ −1 IΓ τr,c, the theorem is valid in this case. r,c ◦ ◦

Case 2 - two points in common: Let 1 be an ellipse and let 2 E E be either an ellipse homothetic to 1 or a straight line such that E 1 2 = O, A . Figure 5 illustrates this case when 2 is a straight line.E ∩E { } E

′′ ′ ′′ C 2 = C 2 C1

A E1

′ C 1

Figure 5. Ellipse 1 and secant line 2. E E 50 Roosevelt Bessoni and Guy Grebot

Let r be the straight line through O parallel to 1 major axis. By E lemma 2.1, the compression τr,c over r with ratio c maps 1 onto ′ ′ E a circle 1 and τr,c ( 2) = 2 is either a circle or a straight line C ′ ′ E C such that 1 2 = O,τr,c(A) . Let Γ be a circle with center O C ∩C { } ′ ′ and let IΓ be the inversion in this circle. IΓ maps 1 and 2 onto C C two concurrent lines according to [8]. The compression τr,c−1 maps concurrent lines onto concurrent lines and it maps Γ onto an ellipse homothetic to 1, as stated by corollary 2.1. E E Since we have IE = τ −1 IΓ τr,c, the theorem is valid in this case. r,c ◦ ◦

Case 3 - no point in common: Let 1 be an ellipse and let 2 be E E either an ellipse homothetic to 1 or a straight line such that 1 2 = E E ∩E . Figure 6 illustrates this case when 2 is a straight line. ∅ E Let r be a straight line parallel to 1 major axis. By lemma 2.1, the E ′ compression τr,c over r with ratio c maps 1 onto a circle 1 and ′ E ′ C ′ τr,c ( 2)= 2 is either a circle or a straight line such that 1 2 = E C ′ C ∩C′′ . According to [8], there is a circle Γ such that IΓ ( 2)= 2 and ∅ ′ ′′ C C ′′ IΓ( 1) = 1 are concentric circles. The images of the circles 2 C ′′ C C and 1 under the compression τ −1 are concentric ellipses. The C r,c image of the circle Γ under the compression τr,c−1 is the desired ellipse of inversion , homothetic to 1. E E

′ ′′ C 1 C 1

′′ C 2

E1 ′ C 2

′′

( C

τ −1 2) c r ,

′′ τr ,c− 1 (C 1)

Figure 6. Ellipse 1 and line 2 with 1 2 = . E E E ∩E ∅

5. CONCLUDING REMARKS The use of plain euclidean geometry arguments has shed light upon the structure of inversions in central conics, which are compositions of compressions and inversions in circles or inversions in equilateral hyperbolae. On inversions in central conics 51

From this structure, the properties of inversions in these central conics fol- low directly from the corresponding properties of inversions in circles and equilateral hyperbolae.

References [1] Childress, N. A., Inversion with respect to the central conics, Mathematics Magazine, 38(3)(1965) 147–149. [2] Heath, T. L. et al., Apollonius of Perga: Treatise on Conic Sections with Introductions Including an Essay on Earlier History on the Subject, Cambridge University Press, Chicago, IL, 1896. [3] Modenov, P. S. and Parkhomenko, A., Euclidean and aﬃne transformations, Published in cooperation with the Survey of Recent East European Mathematical Literature [by] Academic Press, 1966. [4] Neas, S., Anallagmatic curves and inversion about the unit hyperbola, Rose-Hulman Undergraduate Mathematics Journal, 18(1)(2017) 104–122. [5] Patterson, B. C., The origins of the geometric principle of inversion, Isis, 19(1)(1933) 154–180. [6] Ram´ırez, J. L., Inversions in an Ellipse, Forum Geometricorum, 14(2014) 107–115. [7] Ram´ırez, J. L. and Rubiano, G. N., Elliptic inversion of two dimensional objects, International Journal of Geometry, 3(2014) 12–27. [8] Yaglom, I. M., Geometric Transformations IV: Circular Transformations, The Math. Assoc. of America, 2009.

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