Applied Mathematical Sciences, Vol. 8, 2014, no. 20, 963 - 973
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.312725
A Vector Analytic Demonstration
of Pappus’ Construction of an Ellipse
from a Pair of Conjugate Diameters
Brian J. Mc Cartin
Applied Mathematics, Kettering University
1700 University Avenue, Flint, MI 48504-6214 USA
c 2014 Brian J. Mc Cartin. This is an open access article distributed under
Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Abstract
A vector analytic demonstration is provided for Pappus’ construction
of the principal axes of an ellipse from a pair of its conjugate diameters.
This vector analytic approach permits the distinguishing of the major
and minor axes of the ellipse.
Mathematics Subject Classification: 15A72, 51M15, 51N20
Keywords: vector analysis, conjugate diameter, conic section, ellipse
1
Historical Remarks [1]
The discovery of the conic sections in the 4th Century B.C. has been attributed to the Greek geometer Menaechmus, a pupil of Eudoxus and contemporary of Plato. However, his original writings have been lost to us so that
we must rely upon the testimony of Eratosthenes as related by Eutocius in his
commentary on Archimedes in order to justify this attribution. (Proclus also
refers to Menaechmus as the discoverer of the conic sections.)
The historical record markedly improves in the 3rd Century B.C. when we
encounter Apollonius’ monumental Conics [4]. Books I-IV have survived in the
original Greek while Books V-VII have only been preserved in Arabic translation. (In 1710, Edmond Halley attempted a reconstruction of the lost Book
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Brian J. McCartin
VIII.) Among the many jewels of this compendium lies the gem of conjugate
diameters, with which a familiarity is assumed in what follows. Readers desiring suitable background material are directed to the classic tomes by Salmon
[15] and Sommerville [16].
Hofmann and Wieleitner [6] contend that Book I of Apollonius’ Conics contains a construction equivalent to that of the principal axes of an ellipse from
a pair of conjugate diameters. Be that as it may, the first such explicit construction appeared only in the 4th Century A.D. as part of Pappus’ Collection
[5].
Higher geometry had languished since the time of Euclid, Archimedes and
Apollonius and Pappus intended his Collection (Books I-VIII) as a revival of
this Golden Age of Greek Geometry. His effort was to no avail and mathematical history records the 4th Century A.D. as a period of general stagnation
in mathematical development (“the Silver Age of Greek Mathematics”) with
Pappus cast as the last of the great Greek geometers.
At the conclusion of Chapter 17 of Book VIII of The Collection [5, pp.
437-438], Pappus presents his ruler and compass construction (i.e. Euclidean
construction) of the principal axes of an ellipse, in magnitude as well as in
position, given any pair of its conjugate diameters. It is this construction,
detailed in the next section, which is the centerpiece of the present study.
Once obtained, the principal axes may then be used to draw the ellipse by a
variety of methods [2, 7].
Despite the ingenuity of Pappus’ construction, he failed to include its
demonstration in the Collection. Such a proof had to await the genius of
Euler when he published a synthetic demonstration of Pappus’ construction
in the 1753 issue Proceedings of the St. Petersburg Academy of Sciences [3].
This paper had been previously presented by a colleague to the St. Petersburg
Academy in 1750 as Euler had already moved to the Berlin Academy of Sciences in 1741. In typical Euleresque fashion, he not only provides a complete
and lucid demonstration but also presents and demonstrates no fewer than
four entirely different and new constructions for the same problem.
While Euler’s original paper is written in Latin and has not been translated
in toto into English, Pappus’ construction as well as Euler’s synthetic demonstration of it appear in highly condensed form in Salmon [15, p. 173], though
unattributed to either mathematical giant. (See [12] for a more expansive
treatment of Euler’s demonstration.)
This brings us to the subject of the present study. Its purpose is to
present an analytic demonstration of Pappus’ construction as a counterweight
to Euler’s synthetic demonstration. As originally presented, Euler’s synthetic
demonstration does not permit a distinction between the major and minor
axes of the ellipse. This is in stark contrast to the vector analytic approach
adopted herein which easily accommodates such a distinction.
Vector analytic demonstration of Pappus’ construction
2
965
Pappus’ Construction [5, pp. 437-438]
Figure 1: Pappus’ Construction
Proposition 1 (Propostion 14: Pappus’ Construction) Problem: Given
two conjugate diameters of an ellipse, to find the principal axes in both position
and magnitude.
• Given (see Figure 1): Conjugate diameters AB & CD (AB = CD)
intersecting at center E.
• Step 1: Produce EA to H so that EA · AH = DE 2 .
• Step 2: Draw a line, Λ, through A CD.
• Step 3: Bisect EH at K.
• Step 4: Draw KL ⊥ EH meeting Λ at L.
• Step 5: With L as center and LE as radius, draw a circle cutting Λ at
F and G.
• Step 6a: Join EF , then draw AM EF with M lying on EG.
• Step 6b: Join EG, then draw AN EG with N lying on EF .
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Brian J. McCartin
• Step 7a: Construct P on EG such that EP 2 = GE · EM.
• Step 7b: Construct R on EF such that ER2 = F E · EN.
• Step 8: EP and ER are the principal semiaxes.
Pappus unnecessarily makes the assumption that AB < CD. Also, Pappus
erroneously concludes that EP is always the major semiaxis and ER is always
the minor semiaxis. Finally, Pappus omits any demonstration of the validity
of his construction.
3
Λ
Vector Analytic Demonstration
M
G
E
P
D
L
N
A
K
R
F
H
Figure 2: Vector Rendition of Pappus’ Construction
We now paraphrase Pappus’ construction in vector notation thereby providing a new vector analytic proof.
967
Vector analytic demonstration of Pappus’ construction
• Step 1:
EA · AH = EA · (EH − EA) = ED2 ⇒ EH = EA +
ED2
⇒
EA
−→
−→ EA
EA
ED 2
ED2
−−→
−→
EH = EH ·
= EA +
= 1+
·
· EA
2
EA
EA
EA
EA
• Step 2:
(1)
−→
−−→
Λ : EA + t · ED (−∞ < t < ∞)
(2)
−−→ 1 −−→
EK = · EH
2
(3)
• Step 3:
⊥
• Step 4: Defining the orthogonal rotation matrix R =
ED2
−−→ 1 −−→⊥ 1
−−→⊥
EK := R⊥ · EK = · EH = · 1 +
2
2
EA2
0 −1
,
1
0
−→⊥
· EA ,
while
⎛
−−→ −→ ⎞2
−−→ −→
2
ED
ED · EA ⎠ −→⊥
ED
·
EA
−→
−−→
⎝
· EA + · EA .
ED =
2
2 −
EA
EA
EA2
−−→
Thus, the line passing through K and perpendicular to EH is given by
ED2
−−→
−−→⊥ 1
λ : EK + s · EK = · 1 +
2
EA2
ED2
−→ s
· EA + · 1 +
2
EA2
−→⊥
· EA ,
while
⎛
⎛
−−→ −→ ⎞
−−→ −→ ⎞2
2
ED · EA ⎠ −→ ED · EA ⎠ −→⊥
ED
−→ −−→ ⎝
⎝
·EA+t·
·EA .
Λ : EA+t·ED = 1 + t ·
2
2 −
EA
EA
EA2
L is located at the intersection of the lines Λ and λ which is found by
−→
equating coefficients of EA and solving for t = t̂:
ED2 − EA2
−→ −→
−−→
EL = EA + t̂ · ED; t̂ = −−→ −→ .
2ED · EA
(4)
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Brian J. McCartin
• Step 5: The vector equation of the circle centered at L with radius EL
is
−→
−→
(r − EL) · (r − EL) = EL2 .
By Equation (4), this circle will cut the line Λ for t satisfying
−→
−−→ −→ −→
−−→ −→
(EA + t · ED − EL) · (EA + t · ED − EL) = EL2 .
This may be recast as the quadratic equation
−−→ −→
ED 2 · t2 + 2(ED · LA) · t + (LA2 − EL2 ) = 0.
As
−−→ −→
ED · LA = −t̂ · ED2 ; LA2 − EL2 = −ED2 ,
this reduces to
t2 − 2t̂ · t − 1 = 0
with roots
t± = t̂ ±
t̂2 + 1
satisfying the relations
t+ + t− = 2t̂; t+ · t− = −1;
t2+
+
t2−
2
= 4t̂ + 2; t+ − t− = 2 t̂2 + 1.
The corresponding points of intersection F and G are then given by
−→ −→
−−→ −
−→ −→
−−→
EF = EA + t+ · ED; EG = EA + t− · ED.
(5)
−→ −−
→
−→ −−
→
A straightforward computation yields EF · EG = 0 so that EF ⊥ EG.
−−→
−→
−−→
• Step 6a: EM is the orthogonal projection of EA onto EG:
−→ −−
→
−−→ EA · EG −
−→
EM =
· EG.
(6)
2
EG
−−→
−→
−→
• Step 6b: EN is the orthogonal projection of EA onto EF :
−→ −→
−−→ EA · EF −→
· EF .
EN =
EF 2
(7)
• Step 7a: Since AEG is acute, Equation (6) implies that
−→ −
−→
EG · EM = EA · EG.
√
−→ −−→
Thus, EP = EG · EM = EA · EG implies that
−−
→
EG
−→
=
EP = EP ·
EG
−→ −−→
EA · EG −−→
· EG.
EG
(8)
Vector analytic demonstration of Pappus’ construction
969
• Step 7b: Since AEF is acute, Equation (7) implies that
−→ −→
EF · EN = EA · EF .
√
−→ −→
Thus, ER = EF · EN = EA · EF implies that
−→
−→ −→
EF
EA · EF −→
−→
ER = ER ·
=
· EF .
(9)
EF
EF
−→ −→
• Step 8: Since EP ⊥ ER, they are the principal axes of the ellipse
−→
−→
r(t) = cos θ · EP + sin θ · ER (0 ≤ θ < 2π).
(10)
−→
−−→
It remains to be shown that EA and ED are conjugate semidiameters
of this ellipse. This will be accomplished in three stages.
−→
−→
−→
· EP + EN
· ER, A lies
1. Show that A lies on r(t): Since EA = EM
EP
ER
)2 + ( EN
)2 = 1. However,
on r(t) if and only if ( EM
EP
ER
EM
EP
2
+
EN
ER
2
=
−→ −
−
→
−→ −
−
→
−→ −
−
→
−→ −
−
→
EA · EG
EA · EF
EF 2 (EA · EG) + EG2 (EA · EF )
+
=
= 1,
2
2
2
2
EG
EF
EF · EG
since, as a mildly tedious computation will convince, both numera→ −−→
EA · ED)2 ] − (EA2 − ED 2 )2 .
tor and denominator equal 4[(t̂2 + 1) · EA2 · ED2 − (−
−→
−→
2. Show that Λ is tangent to r(t): Since r (t) = − sin θ· EP +cos θ· ER,
and sin θ = EN
at A, a tangent vector at A is given
and cos θ = EM
EP
ER
by
EN −→ EM −→
−→
−−→
T = −
· EP +
· ER = α · EA + β · ED,
ER
EP
where
−→ −→
−→ −−→
ER · EG · EM · EA · EF − EP · EF · EN · EA · EG
=
α=
EP · EF · ER · EG
−→ −→ −→ −
−→
−→ −→ −→ −−
→
(EA · EF ) · (EA · EG) − (EA · EF ) · (EA · EG)
= 0.
EP · EF · ER · EG
−−→
Thus, T ED Λ so that Λ is tangent to r(t).
3. Show that D lies on r(t): By the First Theorem of Apollonius [17,
p. 100], the sum of squares of conjugate semidiameters is constant. (There is a particularly captivating matrix analytic proof
of this theorem appearing in [11]). Thus, if we can show that
−−→
EA2 + ED2 = EP 2 + ER2 then, since we already know that ED
−→ −−→
points in a direction conjugate to EA, ED will also have the correct
−→
length of the semidiameter conjugate to EA so that D must lie on
r(t). Direct computation reveals that
−→ −−→ −→ −→
EP 2 + ER2 − EA2 = EA · EG + EA · EF − EA2 = ED2 .
Q.E.F.
970
4
Brian J. McCartin
Distinguishing Major from Minor Axes
Λ
Λ G
P
L
M
E
A
K
N
R
F
M G
P
KA
E
N
D
D
R
L
H
F
H
H
R
NF
D
E
K
A
M
L
P
F
H
G
Λ
L
AK
G M
P
R
N
D
E
Λ
Figure 3: Obtuse Angle Between Conjugate Semidiameters
The synthetic demonstration of Euler [3, 12] does not allow one to distinguish between the major and minor axes of the ellipse. This is a significant
shortcoming since in orthogonal linear regression the major axis corresponds
to the best linear fit while the minor axis corresponds to the worst linear fit
[8]. However, the vector analytic approach adopted in the present work easily
permits the identification of the major and minor axes.
Given a pair of conjugate diameters, there are four ways to choose the point
A in Pappus’ construction. For each such choice, there are two ways to select
the point D, for a total of eight ways to define the conjugate semidiameters.
Four produce an obtuse angle (see Figure 3) while four produce an acute angle
−→
−−→
(see Figure 4) between the conjugate semidiameters EA and ED.
−→
Observe from these two figures that an obtuse angle yields EP as the
−→
major semiaxis and ER as the minor semiaxis while an acute angle reverses
971
Vector analytic demonstration of Pappus’ construction
D
D
R
F
L
N
E
A
K
M
G
P
M
P
Λ
Λ
H
G
M
P
E
K
A
N
H
L
R
L
H
G
GΛ
H
Λ
NR
F
KA
E
F
D
P
M
L
AK
N
R
F
E
D
Figure 4: Acute Angle Between Conjugate Semidiameters
the semiaxes. A proof follows immediately from the calculation
−→ −−→
−→ −−→
ER2 − EP 2 = (t+ − t− ) · (EA · ED) = 2 t̂2 + 1 · (EA · ED).
Hence,
−→ −−→
−→ −−→
EA · ED < 0 ⇒ EP > ER; EA · ED > 0 ⇒ ER > EP.
5
Concluding Remarks
In the foregoing, an analytic demonstration of Pappus’ construction was
provided that complements the synthetic demonstration of Euler. As should
be abundantly clear to the reader, the approach employed above applied to any
of the constructions surveyed in [6, 13] would likewise yield a vector analytic
demonstration. The importance of such constructions to Applied Mathematics is most easily appreciated by noting the central role of conjugate diameters
972
Brian J. McCartin
in the development of Newtonian mechanics [14]. Furthermore, the GaltonPearson-McCartin Theorem [9, 10] reveals the central role of conjugate diameters in the context of linear regression.
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Vector analytic demonstration of Pappus’ construction
973
[12] B. J. McCartin, On Euler’s Synthetic Demonstration of Pappus’ Construction of an Ellipse from a Pair of Conjugate Diameters, International
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Received: January 1, 2014