A Matrix-Vector Analytic Demonstration of Pappus Construction of an Ellipse from a Pair of Conjugate Diameters

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1 Applied Mathematical Sciences, Vol. 9, 015, no. 14, HIKARI Ltd, A Matrix-Vector Analytic Demonstration of Pappus Construction of an Ellipse from a Pair of Conjugate Diameters Brian J. M c Cartin Applied Mathematics, Kettering University 1700 University Avenue, Flint, MI USA Copyright c 015 Brian J. M c Cartin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A matrix-vector analytic demonstration is provided for Pappus construction of the principal axes of an ellipse from a pair of its conjugate diameters. Mathematics Subject Classification: 15A7, 51M15, 51N0 Keywords: matrix analysis, vector analysis, conjugate diameter, conic section, ellipse 1 Introduction In his 4 th Century A.D. Collection, Pappus presented the first explicit construction of the principal axes of an ellipse given any pair of its conjugate diameters, pp. pp Having omitted any proof of the validity of this construction, Euler supplied its first synthetic demonstration in the 18 th Century 1. An expansive treatment accessible to the modern reader of both Pappus construction as well as Euler s demonstration is available in 8. As a counterweight to this synthetic treatment, a purely vector analytic approach appeared in 9. A distinct advantage of the analytic over the synthetic approach is its ability to readily distinguish between the major and minor axes of the ellipse. This ability is of vital importance in applications such as

2 680 B. J. McCartin orthogonal regression where the major axis corresponds to the best linear fit with the minor axis relegated to the role of worst linear fit 4. In the present paper, a new matrix-vector analytic demonstration of Pappus construction is provided. The foundation for this demonstration will be a matrix analytic theorem (Theorem 1 below) relating the principal axes of an ellipse to any given pair of its conjugate diameters 7. Pappus Construction, pp Figure 1: Pappus Construction At the conclusion of Chapter 17 of Book VIII of his Collection, Pappus presents the following 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. 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.

3 Matrix-vector analytic demonstration of Pappus construction 681 Step : 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. Step 7a: Construct P on EG such that EP = GE EM. Step 7b: Construct R on EF such that ER = F E EN. Step 8: EP and ER are the principal semiaxes. Pappus omits any proof of the validity of this construction but a synthetic demonstration was subsequently provided by Euler 1, 8 which was followed in turn by the vector analytic demonstration of McCartin 9. 3 Matrix Analytic Theorem The following fundamental result 7 will form the cornerstone of the new matrix-vector analytic demonstration to follow. (See Figure.) Theorem 1 (Principal Axes from Given Pair of Conjugate Diameters) a11 a1 Given any pair of linearly independent vectors A 1 =, A =, with A 1 A 0, there is a unique ellipse x T Mx = 1 for which they generate conjugate diameters. Specifically, M = (AA T ) 1 where A = A 1 A. Moreover the principal semiaxes point in the directions a 11 a 1 + a 1 a δ major = a 1 + a A 1 F + A 4F 4det, (1) (A) δ minor = a 11 a 1 + a 1 a a 1 + a A 1 F A 4F 4det, () (A) with corresponding squared magnitudes σmajor = 1 A F + A 4F 4det (A), (3) a 1 a

4 68 B. J. McCartin A δ major σ major A 1 σ minor δ minor Figure : Matrix Analytic Theorem σ minor = 1 where A F = A 1 + A. A F A 4F 4det (A), (4) 4 Vector Analytic Construction We next summarize Pappus construction in vector notation 9. With reference to Figure 3, let EA and ED be the given conjugate semidiameters and define the vectors EF = EA + t + ED; EG = EA + t ED, (5) where t ± = ˆt ± ˆt + 1; ˆt = ED EA ED EA. (6) (A straightforward computation yields EF EG = 0 so that EF EG.) Then, the principal semiaxes of the ellipse are given by EP = EA EG EA EF EG; ER = EG EF EF, (7)

5 Matrix-vector analytic demonstration of Pappus construction 683 Λ G P M E D L K A N R F H Figure 3: Vector Rendition of Pappus Construction so that EP = EA EG, ER = EA EF and EP ER. Specifically, an obtuse angle between the conjugate semidiameters EA and ED results in EP being the major semiaxis and ER the minor semiaxis while an acute angle between them interchanges the semiaxes. This follows immediately from the simple computation ER EP = (t + t ) ( EA ED) = ˆt + 1 ( EA ED). Hence, EA ED < 0 EP > ER; EA ED > 0 ER > EP. 5 Matrix-Vector Analytic Demonstration The identity det (A) = A 1 A (A 1 A ) (8)

6 684 B. J. McCartin implies that ( A 1 + A ) 4det (A) = ( A 1 A ) + 4(A 1 A ), (9) so that Equations (1-4) may be recast as δ major = δ minor = 1 1 a 11 a 1 + a 1 a a 1 + a a 11 a 1 ( A 1 A ) + 4(A 1 A ), (10) a 11 a 1 + a 1 a a 1 + a a 11 a 1 + ( A 1 A ) + 4(A 1 A ), (11) σmajor = 1 A 1 + A + ( A 1 A ) + 4(A 1 A ), (1) A 1 + A ( A 1 A ) + 4(A 1 A ). (13) σminor = 1 Setting A 1 = EA1 EA = EA take the form δ major = δ minor =, A = ED1 ED = ED 1 EA + ED EA 1 ED 1 1 EA + ED EA 1 ED 1 + EA 1 EA + ED 1 ED, Equations (10-13) (ED EA ) + 4( EA ED) EA 1 EA + ED 1 ED (14) (ED EA ) + 4( EA ED) (15) σmajor = 1 EA + ED + (ED EA ) + 4( EA ED), (16) σminor = 1 EA + ED (ED EA ) + 4( EA ED). (17) Turning our attention to Equations (5-7), note that EP = 1 EA + ED sgn ( EA ED) (ED EA ) + 4( EA ED), (18) ER = 1 EA + ED + sgn ( EA ED) (ED EA ) + 4( EA ED). (19),,

7 Matrix-vector analytic demonstration of Pappus construction 685 Defining s major = max (EP, ER ) and s minor = min (EP, ER ), we have s major = 1 s minor = 1 EA + ED + (ED EA ) + 4( EA ED) = σmajor, (0) EA + ED (ED EA ) + 4( EA ED) = σminor. (1) Thus, the square magnitudes EP and ER coincide with those of the principal semiaxes of the ellipse. But what of the directions of EP and ER? Since ER EF = EA + t + ED, EP EG = EA + t ED () with t ± = ED EA ± sgn ( EA ED) (ED EA ) + 4( EA ED) EA, (3) ED we define the corresponding directions d major = EA + ED EA + (ED EA ) + 4( EA ED) EA ED d minor = EA + ED EA (ED EA ) + 4( EA ED) EA ED ED, (4) ED. (5) Since EP ER, all that remains to be shown is that d major δ major as d minor δ minor would thereby follow immediately. A straightforward but rather laborious computation reveals that det d major δ major = 0 so that, indeed, d major δ major. Thus, EP and ER are the principal semiaxes of the ellipse. Q.E.F. 6 Conclusion In the foregoing, a matrix-vector analytic demonstration of Pappus construction was provided that complements both the purely vector analytic demonstration of 9 as well as the synthetic demonstration of Euler 1, 8. As should be abundantly clear to the reader, the application of Theorem 1 above to any of the constructions surveyed in 3, 13 would likewise yield a

8 686 B. J. McCartin matrix-vector analytic demonstration of its validity. Moreover, Theorem 1 may be directly applied to produce such a Euclidean construction 10. The importance of such constructions to Applied Mathematics is most easily appreciated by noting the central role of conjugate diameters in the development of Newtonian mechanics 14. Furthermore, the Galton-Pearson- McCartin Theorem 5, 6 reveals the central role of conjugate diameters in the context of linear regression. Finally, such constructions are directly applicable to assorted problems of linear regression 11, 1. References 1 L. Eulero, Solutio Problematis Geometrici, Novi Commentarii Academiae Scientiarum Petropolitanae, Tom. III (1750/1751), 1753, pp diagram (Tab. IV), T. L. Heath, A History of Greek Mathematics, Volume II: From Aristarchus to Diophantus, Dover, New York, NY, 1981(191). 3 J. E. Hofmann and H. Wieleitner, Zur Geschichte der sog. Rytzschen Achsenkonstruktion einer Ellipse aus einem Paar konjugierter Durchmesser, Nieuw Archief voor Wiskunde (Amsterdam), Series, Part 16, 1930, pp B. J. McCartin, A Geometric Characterization of Linear Regression, Statistics, Vol. 37, No., 003, pp B. J. McCartin, Oblique Linear Least Squares Approximation, Applied Mathematical Sciences, Vol. 4, No. 58, 010, pp B. J. McCartin, Corollary to a Theorem of Oblique Linear Regression, Applied Mathematical Sciences, Vol. 6, No. 57, 01, pp B. J. McCartin, A Matrix Analytic Approach to the Conjugate Diameters of an Ellipse, Applied Mathematical Sciences, Vol. 7, No. 36, 013, pp B. J. McCartin, On Euler s Synthetic Demonstration of Pappus Construction of an Ellipse from a Pair of Conjugate Diameters, International Mathematical Forum, Vol. 8, No., 013, pp

9 Matrix-vector analytic demonstration of Pappus construction B. J. McCartin, A Vector Analytic Demonstration of Pappus Construction of an Ellipse from a Pair of Conjugate Diameters, Applied Mathematical Sciences, Vol. 8, No. 0, 014, pp B. J. McCartin, A Matrix-Vector Analytic Construction for the Pappus- Euler Problem, Applied Mathematical Sciences, Vol. 10, 016, to appear. 11 B. J. McCartin, Geometric Construction of an Ellipse from Its Moments, Applied Mathematical Sciences, Vol. 11, 017, to appear. 1 B. J. McCartin, Geometric Constructions Relating Various Lines of Regression, Applied Mathematical Sciences, Vol. 1, 018, to appear. 13 Z. Nádeník, O konstrukcích os elipsy z jejích sdru zených pr umĕr u (metodicko-historický p ríspĕvek), in M. Ka sparová and Z. Nádeník (Editors), Jan Sobotka ( ) (Czech), Matfyzpress, Praha, 010, pp I. Newton, Principia, Second Edition, University of California Press, Berkeley, CA, 1971(1713). Received: January 1, 015; Published: January 3, 015