Applied Mathematical Sciences, Vol. 10, 2016, no. 3, 127 - 135 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.511704
Geometric Construction of an Ellipse from Its Moments
Brian J. McCartin
Applied Mathematics, Kettering University 1700 University Avenue, Flint, MI 48504-6214 USA
Copyright c 2015 Brian J. McCartin. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduc- tion in any medium, provided the original work is properly cited.
Abstract This paper offers a trio of Euclidean constructions of an ellipse given its first moments and second central moments. The crux of these con- structions lies in the graphical construction of Carlyle [4, p. 99] and Lill [2] for the roots of a quadratic and the Pappus-Euler problem [12] of the construction of the principal axes of an ellipse from a pair of its conjugate diameters.
Mathematics Subject Classification: 51M15, 51N20
Keywords: geometric construction, conic section, ellipse
1 Preliminaries
An ellipse E is uniquely determined by its first moments 1 ZZ 1 ZZ Mx := x dxdy, My := y dxdy, (1) AE E AE E together with its second central moments
ZZ 1 2 Mxx := (x − Mx) dxdy, (2) AE E 1 ZZ Mxy := (x − Mx)(y − My) dxdy, (3) AE E 128 B. J. McCartin
ZZ 1 2 Myy := (y − My) dxdy, (4) AE E where AE is the area of the ellipse. Of course, the inequalities Mxx > 0, 2 Myy > 0 and Mxy < MxxMxx must be satisfied. Specifically [8], the equation of the required ellipse is given by the positive definite quadratic form
" #−1 " # h i Mxx Mxy x − Mx x − Mx y − My = 4, (5) Mxy Myy y − My and the elliptical area may be expressed as
q 2 AE = 4π MxxMyy − Mxy. (6)
The slopes of the principal axes of the ellipse are the roots of the quadratic M − M m2 + xx yy · m − 1 = 0, (7) Mxy so that q 2 2 (Myy − Mxx) + (Myy − Mxx) + 4Mxy mmajor = , (8) 2Mxy
q 2 2 (Myy − Mxx) − (Myy − Mxx) + 4Mxy mminor = , (9) 2Mxy since mmajor has the same sign as Mxy [6]. The corresponding squared magnitudes of the principal semiaxes [15, p. 158] are the roots of the quadratic
2 2 σ − 4(Mxx + Myy) · σ + 16(MxxMyy − Mxy) = 0, (10) so that 2 h q 2 2 i σmajor = 2 (Mxx + Myy) + (Mxx − Myy) + 4Mxy , (11)
2 h q 2 2 i σminor = 2 (Mxx + Myy) − (Mxx − Myy) + 4Mxy . (12) Equation (5) may be expanded thereby yielding
2 2 2 Myy(x − Mx) − 2Mxy(x − Mx)(y − My) + Mxx(y − My) = 4(MxxMyy − Mxy) (13) as the equation of the ellipse. Setting y = My yields the corresponding hori- zontal points of intersection
q 2 Mx ± 2 Mxx − Mxy/Myy,My , (14) Construction of an ellipse from moments 129
while setting x = Mx yields the corresponding vertical points of intersection
q 2 Mx,My ± 2 Myy − Mxy/Mxx . (15)
The points of vertical and horizontal tangency are [7], respectively,
! q Mxy Mxy q Mx ± 2 Mxx,My ± 2√ ; Mx ± 2q ,My ± 2 Myy . (16) Mxx Myy
A 2
A 1
Figure 1: Conjugate Semidiameters
With reference to Figure 1, recall that two semidiameters are conjugate to one another if they are parallel to each others tangents [8]. For this to occur, it is necessary and sufficient that their slopes m and s satisfy the symmetric conjugacy condition: [7]
Mxx · (m · s) − Mxy · (m + s) + Myy = 0. (17)
Thus, the semidiameter to a point of vertical/horizontal tangency is conjugate to the semidiameter to a point of vertical/horizontal intersection, respectively. (See Figures 4/5, Left.) In the succeeding sections, a trio of straightedge-compass constructions of an ellipse, E, given its moments, hMx,My,Mxx,Mxy,Myyi, will be developed. Strictly speaking, such a Euclidean construction is not possible for the ellipse. 130 B. J. McCartin
However, given the principal axes of the ellipse, several methods are available for drawing any number of points on the ellipse [3]. Thus, the phrase “construct the ellipse” will be interpreted as “construct the principal axes of the ellipse”. Certain primitive arithmetic/algebraic Euclidean constructions [1, pp. 121- 122], as next described, will be required. Proposition 1 (Primitive Constructions) Given two line segments of non- zero lengths a√and b, one may construct line segments of length a + b, a − b, a · b, a/b and a using only straightedge and compass.
2 The Carlyle-Lill Construction
m major
(0,1) m minor
(M −M )/M yy xx xy m m minor major (M ,M ) x y
−1 ((M −M )/M ,−1) yy xx xy
Figure 2: Determination of Axes Directions √ Because Equations (8-9) involve only the operators h+, −, ·, /, ·i, applica- tion of Proposition 1 would provide an immediate construction of the directions of the principal axes. Furthermore, since Equations (11-12) also involve only this same set of operators, an additional application would yield the squared magnitudes of the principal semiaxes thereby leading to their complete con- struction. Clearly, an approach less reminiscent of a sledgehammer is to be preferred. Instead, the following beautiful construction due to Carlyle [4, p. 99] and Lill [2, pp. 355-356] will be invoked. Proposition 2 (Graphical Solution of a Quadratic Equation) Draw a circle having as diameter the line segment joining (0, 1) and (a, b). The ab- scissae of the points of intersection of this circle with the x-axis are the roots of the quadratic x2 − ax + b = 0. Construction of an ellipse from moments 131
(4(M +M ),16(M M −M2 )) xx yy xx yy xy
σ major
σ minor
(0,1) + + σ2 σ2 minor major
Figure 3: Determination of Semiaxes Magnitudes
A double application of Proposition 2, in concert with Equations (7) and (10), produces the following far more elegant construction.
Construction 1 (Double Carlyle-Lill Construction) Given the first and second moments hMx,My,Mxx,Mxy,Myyi:
1. According to Equation (7), apply the Carlyle-Lill construction, Propo- sition 2, with (a, b) = ( Myy−Mxx , −1) to construct the directions of the Mxy principal axes. (See Figure 2.)
2. According to Equation (10), apply the Carlyle-Lill construction, Propo- 2 sition 2, with (a, b) = (4(Mxx + Myy), 16(MxxMyy − Mxy)) to construct the squared magnitudes of the principal semiaxes. (See Figure 3, Left.) √ 3. Apply the ·-operator, Proposition 1, to these squared magnitudes in or- der to construct the magnitudes, {σmajor, σminor}, of the principal semi- axes emanating from the centroid, (Mx,My). (See Figure 3, Right.)
4. From the principal semiaxes, {~σmajor, ~σminor}, a variety of methods [3] may now be employed to construct any number of points lying on the ellipse
E : ~r(t) = cos θ · ~σmajor + sin θ · ~σminor (0 ≤ θ < 2π). 132 B. J. McCartin
Figure 4: Vertical Tangent Construction 3 Pappus-Euler Problem
The following problem was first formulated and solved by Pappus in the 4th Century, although Euler gave the first proof of its validity only in the 18th Century [9]. Proposition 3 (Pappus-Euler Problem) Given two conjugate diameters of an ellipse, to find the principal axes in both position and magnitude. As Euler’s demonstration of Pappus’ construction was purely synthetic in na- ture, McCartin subsequently provided both a purely vector analytic demon- stration [10] as well as an independent matrix-vector analytic demonstration [11]. Since that time, many alternative solutions have been devised [5, 14]. Most recently, McCartin has provided a matrix-vector analytic construction [12] which will next be utilized, in tandem with some prior observations, to provide a pair of related constructions of an ellipse from its given moments. As previously noted, the semidiameter of a point of vertical intersection is conjugate to the semidiameter to a point of vertical tangency. Thus, by Equations (15) and (16), the vectors
q q 2 Mxy h0, 2 Myy − Mxy/Mxxi; h2 Mxx, 2√ i (18) Mxx define a pair of conjugate semidiameters emanating from the centroid, (Mx,My), which may be utilized in the Pappus-Euler problem, Proposition 3, to construct the principal axes of the ellipse. (See Figure 4.) Construction of an ellipse from moments 133
Figure 5: Horizontal Tangent Construction
Construction 2 (Vertical Tangent Construction) Given the first and sec- ond moments hMx,My,Mxx,Mxy,Myyi, apply the McCartin algorithm [12] to the pair of conjugate semidiameters
q q 2 Mxy h0, 2 Myy − Mxy/Mxxi; h2 Mxx, 2√ i Mxx thereby constructing the principal axes of the ellipse in direction and magnitude emanating from the centroid, (Mx,My).
Likewise, the semidiameter of a point of horizontal intersection is conjugate to the semidiameter to a point of horizontal tangency. Thus, by Equations (14) and (16), the vectors
q q 2 Mxy h2 Myy − Mxy/Mxx, 0i; h2q , 2 Myyi (19) Myy define a pair of conjugate semidiameters emanating from the centroid, (Mx,My), which may be utilized in the Pappus-Euler problem, Proposition 3, to construct the principal axes of the ellipse. (See Figure 5.)
Construction 3 (Horizontal Tangent Construction) Given the first and second moments hMx,My,Mxx,Mxy,Myyi, apply the McCartin algorithm [12] to the pair of conjugate semidiameters
q q 2 Mxy h2 Myy − Mxy/Mxx, 0i; h2q , 2 Myyi Myy 134 B. J. McCartin thereby constructing the principal axes of the ellipse in direction and magnitude emanating from the centroid, (Mx,My).
4 Postscript
In the foregoing, a trio of Euclidean constructions for an ellipse given its first moments and second central moments was developed. The first such construction relied heavily upon the Carlyle-Lill graphical construction of the roots of a quadratic equation. The remaining constructions hinged upon the many available solutions to the Pappus-Euler problem for the construction of the principal axes of an ellipse given any pair of its conjugate diameters [5, 9, 10, 11, 12, 14]. While the present paper was focused on continuous distributions of mass, given the appropriate first and second moments, the very same constructions are applicable to the construction of the “concentration ellipse” [6] associated with a discrete distribution of points. Furthermore, this observation permits the unified construction of various lines of regression [13].
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Received: December 1, 2015; Published: January 14, 2016