INTERNATIONAL JOURNAL OF GEOMETRYVol. 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 ofan inversion in an ellipse or in a hyperbola. By this formulation, results oninversions 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 thebases of inversive geometry in a text dated from 1824 and published after hisdeath by Butzberger. In 1965, the mathematician Noel Childress [1] extendedthe concept of circular inversion to central conics, ellipse or hyperbola. In2014, Ramirez [6][7] showed other properties concerning inversion in ellipsesand Neas [4], in 2017, looked at anallagmatic curves under inversion inhyperbolae.

Inversion in circles is a geometrical topic which is usually developpedwithout analytic geometry [8]. Strangely enough, we could not find anywork concerning inversion in central conics using plain euclidean geometryarguments. In the studied publications the basic framework is analyticalgeometry. Also, as far as we could reach, the structure of an inversion in acentral conic is not mentioned in the cited works and it seems to us that theanalytical framework tends to hide this property.

In this communication, we use plain euclidean geometry to analyse thestructure of an inversion in an ellipse or in a hyperbola. We show thatan inversion in a central conic is given by the composition of compressionsand an inversion in a circle or in an equilateral hyperbola; in this sense, aninversion in a central conic is just an affine deformation of an inversion in acircle or in an equilateral hyperbola.

————————————–Keywords and phrases: Inversion, compression, inversion in central

conics.(2010)Mathematics Subject Classification: 51M04, 51M05Received: 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 ofinversions in ellipses which admit this circle as an auxiliary circle. In thesame way, to each equilateral hyperbola corresponds a family of inversionsin hyperbolae with same auxiliary circle.

In the next section, we define inversion and compression and we stateand prove an important result about compressions in parallel lines. Weshow the main result of this communication in section 3. Section 4 treatsthe immediate consequences of the main theorem as well as the proof of aresult 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 thealgebraic measurement (signed length) of such segment will be indicated byAB. 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 inver-

sion in the central conic is the transformation I of Π which associates,

to each point P 6= O in Π, a point P ′ on the ray−−→OP , such that

(1) OP OP ′ = OQ2,

where Q ∈ ∩ −−→OP .

It follows directly from this definition that: a point P ∈ Π \ {O} has an

image under the inversion in , I(P ), only if the ray−−→OP intercepts ; I

is a 1 − 1 mapping; if is an ellipse or a circle, the inversion in , I, isdefined for all points in Π \ {O}; if is a circle of center O, OQ is its radiusand the righthand side of (1) is constant; I(P ) = P if, and only if, P ∈ ;if P ′ = I(P ) then P = I(P ′); a straight line through O that intercepts is invariant under I.

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] andreferences therein.

It is to be noted that the treatment of inversions in hyperbolae is moredelicate than for inversions in ellipses. This is due to the fact that the defi-nition allows transforming only points that lie in the region limited by theassymptots and which contains the vertices. Although this remark does notaffect the results presented below, it contributes to the lack of symmetry be-tween the properties of inversions in hyperbolae and properties of inversionsin ellipses.

As will be seen in the next section, the main result of this communicationexpresses that inversions in ellipses and hyperbolae are related to inversionsin circles and equilateral hyperbolae, respectively, via a compression.

A compression is an affine transformation defined as follows [3].

Definition 2.2. Let c 6= 0 be a real number and let l be a straight line in

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

−−−→PM ′ = c

−−→PM,

where P ∈ l is the orthogonal projection of M over l.

Lemma 2.1. There is a compression that maps an ellipse E with center Oonto a circle centered in O.

Proof. The given ellipse E is defined by PF1 + PF2 = a, for all P ∈ E ,where F1 and F2 are its foci and F1F2 = c. Let τ be the compression overthe focal axis of E with coefficient k. Then, for every P ∈ E , P ′ = τ(P ) is

such that−−−→P1P

′ = k−−→P1P , where P1 is the orthogonal projection of P over

the focal axis of the ellipse E . Figure 1 illustrates these settings. From

OF1 F2 B

E

P 0

P1

P

A

Figure 1. Illustration of Lemma 2.1.

(

PF1 + PF2

)2= a2, we get

(

2PF1 PF2

)2=

[

a2 −(

PF12+ PF2

2)]2

,

4PF12PF2

2= a4 − 2 a2

(

PF12+ PF2

2)

+(

PF12+ PF2

2)2

,

PF14+ PF2

4 − 2PF12PF2

2= 2 a2

(

PF12+ PF2

2)

− a4,

(

PF12 − PF2

2)2

= 2 a2(

PF12+ PF2

2)

− a4.(2)

On inversions in central conics 43

Since PP1 ⊥ F1F2 and the points P1, F1 and F2 are collinear, we have(

PF12 − PF2

2)2

=(

P1F12 − P1F2

2)2

=(

P1F12 − P1F2

2)2

=(

P1F1 − P1F2

)2 (

P1F1 + P1F2

)2,

P1F1 = P1O +OF1,

P1F2 = P1O +OF2.

So equation (2) can be written as

4 c2 OP12

= 2a2(

c2

2+ 2.OP1

2+ 2.P1P

2)

− a4,

4(

c2 − a2)

OP12

= 4a2 P1P2+ a2

(

c2 − a2)

.(3)

As OP12= OP ′

2 − P1P ′2= OP ′

2 − k2 P1P2, equation (3) becomes

OP ′2 − k2 P1P

2=

a2

(c2 − a2)P1P

2+

a2

4,

OP ′2 − a2

4= P1P

2(

k2 +a2

c2 − a2

)

.

Since a > c, we can put k =a√

a2 − c2and we obtain OP ′ =

a

2. This is

what we claimed for, i.e. that the compression over the focal axis of E and

ratio k =a√

a2 − c2takes the point P ∈ E onto the point P ′ on the circle of

radiusa

2centered in O. �

An ellipse with center O has two auxiliary circles centered in O: one hasradius equal to the semi-major axis and the other has radius equal to thesemi-minor axis of the ellipse. In the same way as the preceeding lemmaexhibits the compression that takes the ellipse onto the outer auxiliary cir-

cle, it can be shown that the compression with ratio k =

√a2 − c2

aand axis

on the perpendicular to the focal axis through O, takes the ellipse onto theinner auxiliary circle.

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

equilateral hyperbola.

Proof. Let t1 and t2 be the assymptots of H and let r be its focal axis. Letτc be the compression over r with ratio c such that t1 and t2 are mappedonto s1 and s2, respectively, where s1 ⊥ s2.

Let A1 and A2 be the vertices of H and let O be its center. Accordingto [2], proposition 31, the hyperbola H′ with vertices A1 and A2 and withassymptots s1 and s2 is uniquely defined, as illustrated in Figure 2.

If p is the parameter of H, we have [2]

LL′2

= pA1A2,

where L ∈ t1, L′ ∈ t2 and A1 ∈ LL′.

44 Roosevelt Bessoni and Guy Grebot

H

r

t1

t2

H 0s1

s2

L

L 0

L 1

L 01

O

A1A2 V

Q

Q0

Figure 2. Illustration of Lemma 2.2.

Let L1 = τc(L) and L′1 = τc(L

′). Then

L1L′1

2= c2 LL′

2

= c2pA1A2

which means that the parameter of H′ is given by p′ = c2p. Using thenotation of [2], the ordinate Q′ ∈ H′ at the point V on the line A1A2 is

Q′V2

= p′A1V A2V

A1A2

= c2pA1V A2V

A1A2

= c2 QV2,

where Q ∈ H is the ordinate relative to V . We then have

Q′V = cQV .

Since V , Q and Q′ are colinear, we can state that−−→Q′V = c

−−→QV ,

which means that

H′ = τc (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

k1, k2 ∈ R \ {0}. The compressions τl1,k1 and τl2,k2 satisfy

τl1,k1 ◦ τl2,k2 = Tu ◦ τl1,k1k2 ,(4)

= Tv ◦ τl2,k1k2 ,(5)

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

−−−→P2M

′ =

k2−−−→P2M and

−−−→P1M

′′ = k1−−−→P1M

′, where P1 and P2 are the orthogonal projec-tions of M over l2 and of M ′ over l1, respectively. From these, we obtain

−−−→P1M

′′ = k1−−−→P1M

′

= k1

(−−−→P2M

′ +−−−→P1P2

)

= k1

(

k2−−−→P2M +

−−−→P1P2

)

= k1k2

(−−−→P1M −−−−→

P1P2

)

+ k1−−−→P1P2

which gives−−−→P1M

′′ = k1k2−−−→P1M + k1(1− k2)

−−−→P1P2.

We also have

−−−→P1M

′′ = k1

(

k2−−−→P2M +

−−−→P1P2

)

−−−→P2M

′′ = k1

(

k2−−−→P2M +

−−−→P1P2

)

−−−−→P1P2

and so−−−→P2M

′′ = 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 following 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 hy-

perbolae;

(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 the-orem.

Theorem 3.1. Let V be a central conic with major axis a and minor axis

b. The inversion with respect to this conic is expressed by the product

(6) IV = τ 1

c

◦ I ◦ τc,

where τc is the compression over V’s focal axis with ratio c =a

b, and I is

either the inversion in the circle concentric to V with radiusa

2, if V is an

ellipse, or the inversion in the equilateral hyperbola with same focal axis and

major axis a as V, in case V is a hyperbola.

We shall call , such that IV = τ 1

c

◦ I ◦ τc, the associated curve to 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

OP OP ′ = OH2

whereH is the point of intersection between the ray−−→OP and V. Let τc(H) =

H ′ ∈ . Figure 3 illustrates these settings.

V

O

◌

P ′

P0

Q′

H

H ′

P

P ′ ′

Figure 3. Illustration of Theorem 3.1 in the case V is an ellipse.

On inversions in central conics 47

Since a compression is an affine transformation, it preserves ratios ofcolinear segments. Thus

OP

OP ′=

OP ′′

OQ′and

OH

OP ′=

OH ′

OQ′.

From these equations, we obtain

(7) OP ′′OQ′ =OP OQ′

2

OP ′= OH ′

2.

Again, using the definition of inverse in a central conic, equation (7) meansthat Q′ = I(P ′′) and so

τc−1 ◦ I ◦ τc(P ) = τc−1 ◦ I(P ′′)

= τc−1(Q′)

= P ′ = IV(P )

which completes the proof that

IV = τc−1 ◦ I ◦ τc.�

4. CONSEQUENCES OF THE MAIN THEOREM

From Theorem 3.1 we see that each circle corresponds to a familly ofinversions in concentric ellipses Vi. Likewise, each equilateral hyperbola corresponds to a familly of inversions in concentric hyperbolae Vi. Two mem-bers of such a familly, which are inversions in concentric ellipses/hyperbolaeV1 and V2, satisfy

τl1,c1 ◦ IV1◦ τl1, 1

c1

= I = τl2,c2 ◦ IV2◦ τl2, 1

c2

,

where τli,ci , i = 1, 2 is a compression of ratio ci over the axis li. Consequently,we have

(8) IV1= τl1, 1

c1

◦ τl2,c2 ◦ IV2◦ τl2, 1

c2

◦ τl1,c1 .

Since τl2, 1

c2

◦ τl1,c1(V1) = V2, it is clear that the composition of compressions

φ = τl2, 1

c2

◦ τl1,c1 is the composition of a rotation about the center and a

compression of ratioc1c2.

In the case of inversions in hyperbolae, the families are disjoint. But, inorder to have disjoint families of inversions in ellipses, we must consider thecircles to be only circunscribed auxiliary circles or only inscribed auxiliarycircles.

The properties of inversions in ellipses cited in [1] and [6] are easily seento be valid, since a compression preserves straight lines and maps circlesonto ellipses. For example, the image of a straight line l by the inversion inan ellipse V1 would be the image, under the compression τ 1

c

, of the image

of τc(l) by the inversion in the associated circle to V1. As such, it has to bean homothetic ellipse to V1 in the case where l does not pass through thecentre, or a straight line otherwise.

48 Roosevelt Bessoni and Guy Grebot

Another interesting example, cited in [7], concerns the transformation ofa pair of perpendicular straight lines under inversion in an ellipse V1 withcenter O. Suppose the point of intersection A, between the perpendicularlines s and t, is distinct from the center O of inversion and that O /∈ s. Thenthe images of τc(s) and τc(t) by the inversion in the associated circle to V1

will be either

(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 trans-formation under τ 1

c

, the tangents at O are mapped onto perpendicular lines

through O and circles are mapped onto ellipses homothetic to V1. As theresult of the inversion we get an ellipse and a straight line or two homotheticellipses that intercept perpendicularly at O.

Another consequence of Theorem 3.1 concerns anallagmatic curves underinversion in a central conic V.Corollary 4.1. A curve C is invariant by the inversion in a central conic

V if, and only if, τl,c(C) is invariant by the inversion in , the associated

curve to V.Proof.

IV(C) = τl, 1c

◦ I ◦ τl,c(C) = C ⇔ τl,c(C) = I ◦ τl,c(C).�

We end this communication by proving a result about inversion in ellipseswhich is a new complement to the properties already published in the lit-erature. It is adapted from a theorem cited in [8] that expresses that linesand circles can be mapped into lines or concentric circles by an inversion ina circle.

Theorem 4.1. Under an inversion in an ellipse E, any two ellipses homo-

thetic to E, or a line and an ellipse homothetic to E, can be transformed into

concurrent lines, parallel lines, or two concentric ellipses.

Proof.The proof will be developped by considering three cases, dependingon whether the given objects (i.e. two ellipses or one ellipse and a line) havea single point in common, two points in common or no point in common.

Case 1 - a single point in common: Let E1 be an ellipse and letE2 be either an ellipse homothetic to E1 or a straight line such thatE1 ∩ E2 = {O}. Figure 4 illustrates this case when E2 is a straightline.

Let r be the straight line through O parallel to E1 major axis. Bylemma 2.1, the compression τr,c over r with ratio c maps E1 onto acircle C′

1 and τr,c (E2) = C′2 is either a circle or a straight line such

that C′1 ∩ C′

2 = {O}. Let Γ be a circle with center O and let IΓ bethe inversion in this circle. IΓ maps C′

1 and C′2 onto two parallel

lines C′′1 and C′′

2, respectively, according to [8]. The compressionτr,c−1 maps parallel lines onto parallel lines and it maps Γ onto anellipse E homothetic to E1, from corollary 2.1.

On inversions in central conics 49

E1C′ 1

O

E2C′ ′ 2 = C′ 2 C′ ′ 1

r

Figure 4. Ellipse E1 and tangent line E2.

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

Case 2 - two points in common: Let E1 be an ellipse and let E2be either an ellipse homothetic to E1 or a straight line such thatE1 ∩E2 = {O,A}. Figure 5 illustrates this case when E2 is a straightline.

E1

C′ 1

O

A

E2

C′ ′ 2 = C′ 2C ′ ′1

Figure 5. Ellipse E1 and secant line E2.

50 Roosevelt Bessoni and Guy Grebot

Let r be the straight line through O parallel to E1 major axis. Bylemma 2.1, the compression τr,c over r with ratio c maps E1 ontoa circle C′

1 and τr,c (E2) = C′2 is either a circle or a straight line

such that C′1 ∩ C′

2 = {O, τr,c(A)}. Let Γ be a circle with center Oand let IΓ be the inversion in this circle. IΓ maps C′

1 and C′2 onto

two concurrent lines according to [8]. The compression τr,c−1 mapsconcurrent lines onto concurrent lines and it maps Γ onto an ellipseE homothetic to E1, as stated by corollary 2.1.Since we have IE = τr,c−1 ◦ IΓ ◦ τr,c, the theorem is valid in this case.

Case 3 - no point in common: Let E1 be an ellipse and let E2 beeither an ellipse homothetic to E1 or a straight line such that E1∩E2 =∅. Figure 6 illustrates this case when E2 is a straight line.Let r be a straight line parallel to E1 major axis. By lemma 2.1, thecompression τr,c over r with ratio c maps E1 onto a circle C′

1 andτr,c (E2) = C′

2 is either a circle or a straight line such that C′1∩C′

2 =∅. According to [8], there is a circle Γ such that IΓ (C′

2) = C′′2 and

IΓ(C′1) = C′′

1 are concentric circles. The images of the circles C′′2

and C′′1 under the compression τr,c−1 are concentric ellipses. The

image of the circle Γ under the compression τr,c−1 is the desiredellipse of inversion E , homothetic to E1. �

E2

E1

r

C′ ′ 1

C′ ′ 2

C′ 2

C′ 1

τr ,c− 1 (C′ ′2)

τr ,c− 1 (C′ ′ 1)

Figure 6. Ellipse E1 and line E2 with E1 ∩ E2 = ∅.

5. CONCLUDING REMARKS

The use of plain euclidean geometry arguments has shed light upon thestructure of inversions in central conics, which are compositions of com-pressions 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 andequilateral 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, Publishedin 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-HulmanUndergraduate 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] Ramırez, J. L., Inversions in an Ellipse, Forum Geometricorum, 14(2014) 107–115.[7] Ramırez, 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.

DEPARTMENT OF MATHEMATICSUNIVERSITY OF BRASILIABRASILIA, 70910-900 DF, BRAZILE-mail address: bessoni@gmail.com

DEPARTMENT OF MATHEMATICSUNIVERSITY OF BRASILIABRASILIA, 70910-900 DF, BRAZILE-mail address: guy@mat.unb.br