The Steiner in-ellipse and moments of inertia of triangles and other polygons

G. Keady and P. Scales

1 The triangle results

Given a triangle

T

in the plane, there is an inscribed ellipse, tangent to each side at the midpoint of the side. This ellipse is usually called the Steiner in-ellipse. There is a substantial popular literature on Steiner ellipses including books such as [2] (where Problem 98 is particularly relevant) and articles on the internet including in wikipedia and in mathworld. Recent articles in publications by MAA, [6,7,9] and elsewhere, e.g.
http://demonstrations.wolfram.com/MardensTheorem

have called attention to it, and some of its applications. The aim of this note is to alert readers that there are mechanics applications, with substantial results noted in the nineteenth century literature: see [10].

Denote the vertices of the triangle

T

z_{1}

z_{2}

z_{3}

, and define its centroid

z_{c} = (z_{1} + z_{2} + z_{3}) / 3

z_{c}

is the centre of the Steiner in-ellipse. Considering the points in 2-dimensional real Euclidean space we may record the coordinates in column vectors

z_{j} = (x_{j, 1}, x_{j, 2})^{T}

, the superscript

^{T}

denoting transpose. The structure of the equation for the Steiner in-ellipse is, for some positive definite matrix

A

(z - z_{c})^{T} A (z - z_{c}) = 1 .

(1)

The entries of

A

are functions of the vertices, readily computed by any Computer Algebra System (CAS).

Consider mass distributed uniformly over

T

. Define a second-moment tensor by

I_{area} = [\begin{matrix} I_{11} & I_{12} \\ I_{12} & I_{22} \end{matrix}]

where

I_{ij} = \int_{T} (X_{i} - x_{c, i}) (X_{j} - x_{c, j}) {dX}_{1} {dX}_{2} .

(2)

Define also a rotation

R

and a reflection

S

R = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}], S = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] .

The usual area moment of inertia tensor is

I_{M . I .} = R I_{area} R^{T}

, though there are some authors that use

S

where

R

is used here. (The results in the mechanics literature, e.g. [10], concern the moment of inertia tensor rather than our second-moment tensor.)

It is convenient to define another second moment tensor,

I_{point}

, this time associated with a triangle with its mass located at the vertices, an equal mass at each vertex. It happens that

I_{point}

is just a scalar multiple of

I_{area}

Once again, these second-moment tensors are positive definite. It can be checked (easily with a CAS doing the routine calculations) that matrices

A

and

I_{area}

commute,

A I_{area} = I_{area} A .

This leads to the following.

THEOREM. The principal axes of the second-moment tensor for the triangle coincide with the directions of the principal axes of the Steiner in-ellipse.

A proof follows from using the fact that commuting diagonalizable matrices are simultaneously diagonalizable, i.e. share eigenvectors. (See [5], items 1.3.19 and 2.3.3.)

Rather more is true. It can be shown that

R I_{area} R^{T}

is just a scalar multiple of

A

The result, with a different proof, is at page 26 of [10]. There is a substantial literature on moments of inertia of triangles including [3,4,11].

2 Polygons and polynomials

2.1 Moments of inertia for systems of equal point masses

Consider next a polynomial of degree

n

, and, for the sake of definiteness, suppose that all the roots are distinct. Suppose also that the sum of the roots is zero, which is only supposed to save some writing in some future formulae:

p (z) = Π_{i = 1}^{n} (z - z_{i}), \sum_{i = 1}^{n} z_{i} = 0 .

(3)

With polynomial

p (z)

we associate a polygon with

n

vertices. With its derivative

p' (z)

we can associate, similarly, a

(n - 1)

-gon.

The set of vertices is the important item, defining both the polynomial and, possibly less importantly, the polygon. Below we define the second-moment matrix

I_{point}

for any

n

-gon with equal masses located at the vertices and taking the moments about the vertex-centroid. The connection with the Steiner in-ellipse is that if one repeatedly differentiates the polynomial until one reaches a cubic, then finds the Steiner in-ellipse associated with the triangle defined from the roots of the cubic, the principal axes of

I_{point}

are the axes of the Steiner in-ellipse. Intermediate results on the way through to this are also memorable.

The derivative, of course is

p' (z) = n Π_{j = 1}^{n - 1} (z - z_{j}^{'}), \sum_{j = 1}^{n - 1} z_{j}^{'} = 0 .

(4)

The centroid of the

(n - 1)

-gon associated with

p'

and the

n

-gon associated with

p

coincide. There is a nice picture, associated with illustrating the Gauss-Lucas Theorem, at the Mathematica demonstrations site,
http://demonstrations.wolfram.com/LucasGaussTheorem
We will return to this later.

The previous sentence refers to a fact about the first moments. Consider next the second-moment matrices:

I_{point} = [\begin{matrix} \sum x_{i}^{2} & \sum x_{i} y_{i} \\ \sum x_{i} y_{i} & \sum y_{i}^{2} \end{matrix}], I'_{point} = [\begin{matrix} \sum (x'_{j})^{2} & \sum x'_{j} y'_{j} \\ \sum x'_{j} y'_{j} & \sum (y'_{j})^{2} \end{matrix}] .

We will show that these matrices have the same eigenvectors, and will do this by showing that the matrices commute.

The commutator

(I I' - I' I)

of two symmetric matrices

I

and

I'

is necessarily skew symmetric and it is the zero matrix iff

I'_{12} (I_{11} - I_{22}) - I_{12} (I'_{11} - I'_{22}) = 0 .

Using the specific entries for our matrices

\begin{matrix} (I I' - I' I) = 0 & \Leftrightarrow & c : = \sum x'_{j} y'_{j} \sum (x_{i}^{2} - y_{i}^{2}) - \sum x_{i} y_{i} \sum ((x'_{j})^{2} - (y'_{j})^{2}) = 0 \\ \Leftrightarrow & c : = Im (\overline{\sum (z'_{j})^{2}} \sum z_{i}^{2}) = 0 . & (5) \end{matrix}

We now return to considering the polynomials

p

and

p'

. Considering the coefficient of

z^{n - 2}

p

and of

z^{n - 3}

p'

, we have

n \sum_{j, k j \neq k} z'_{j} z'_{k} = (n - 2) \sum_{i, k i \neq k} z_{i} z'_{k} .

From this, on using the centroid conditions, first moment conditions, of equations (3,4) we have

\sum (z'_{j})^{2} = \frac{(n - 2)}{n} \sum z_{i}^{2} .

Using this in the expression

c

of equation (5), we have that

\begin{matrix} c & = & \frac{(n - 2)}{n} Im (| \sum z_{i}^{2} |^{2}) \\ = & 0 . \end{matrix}

This establishes that the matrices

I_{point}

and

I'_{point}

commute and hence share eigenvectors.

Now consider the picture, showing not just the

n

-gon and the

(n - 1)

-gon associated with

p

and

p'

respectively, but also the roots for the higher order derivatives. See the picture associated with the Gauss-Lucas Theorem, at the Mathematica demonstrations site. Eventually we get to a cubic polynomial, a triangle, and its Steiner in-ellipsoid. The axes of this ellipsoid are the principal axes for

I_{point}

, and for

I_{point}^{(k)}

for any, e.g.

k

-th, derivative's set of points.

2.2 Area moments of inertia for polygons

We define a polygon as an ordered list of vertices, and may suppose that the coordinates of the vertices are represented as pairs of reals or, as

(z_{j})_{j = 1}^{n}

complex numbers. The generalization of the preceding formula, equation (2) for

I_{area} (z_{c})

from triangles

T

to general polygons merely requires us to replace the region of integration

T

by the polygon

P

, and it is useful to note that

z_{c}

could be, in this more general context, any point, and we prefer to indicate the centre about which the moments are taken by

z_{*}

. (In engineering applications it is likely that crossed polygons and so on would not occur, but they are not a problem in the following.) It is well known that there are multivariate polynomial expressions for the entries of

I_{area}

in terms of the coordinates of the vertices: see [1], for example. The formulae have been coded in many languages and are, for example, in Mathematica's Structural Mechanics Pack.

The Parallel Axes Theorem is a well-known result indicating how the moment matrices

I_{area} (z_{*})

change as

z_{*}

changes. The open question we have is whether there might be some simple formula for the point we take moments about so that the eigenvectors of

I_{area} (z_{*})

align with the axes of the Steiner in-ellipse of the preceding subsection. For triangles, the result is mentioned in §1, namely that the area centroid, which coincides with the vertex average, is such a point. For quadrilaterals, the area centroid need no longer coincide with the vertex average, and it happens that if we take moments about the vertex average the principal axes coincide with the axes of the Steiner in-ellipse (and, in fact, denoting the vertex average by

z_{VA}

, we have that the matrix

I_{area} (z_{VA})

is a scalar multiple of

I_{point})

.) We have no results for polygons with larger numbers of sides.

3 Tetrahedra and higher dimensions

There are generalizations to higher dimensions, presumably to

n

-dimensions, though the mechanics literature stops at

n = 3

. Let

T

be a tetrahedron in Euclidean 3-space. Define the Steiner in-ellipsoid to be the inscribed ellipsoid tangent to each face at the face's centroid. The centre of the Steiner ellipsoid is located at the centroid of the tetrahedron. (Steiner ellipsoids have already been defined in earlier literature, from the 1940s and possibly earlier. As in the plane case, these usually refer to circum-ellipsoids.) Once again, the equation of the ellipsoid can be written in the form of equation (1), with, now,

A

3 \times 3

matrix. An analogue, for this three dimensional situation, of the theorem above is available in [10], pp28-30, i.e. that the principal axes of inertia correspond to the eigenvectors of

A

. (See also [8].) The CAS-based proof techniques we used also generalize: it is easy to verify, with the obvious definitions of the

3 \times 3

matrices,

I_{point} A = A I_{point}

References

[1]: S.F. Bockman, Generalizing the formula for areas of polygons to moments, Amer. Math. Monthly 96 (Feb 1989), 131-132.
[2]: H. Dorrie, 100 Great Problems of Elementary Mathematics, Their History and Solution, translated (from German to English) by D. Antin, Dover, New York, 1965.
[3]: N.M. Gibbins, The moment of inertia of a triangle about any line in its plane, Math. Gazette 12 (1925), 392.
[4]: W. Hope-Jones, Moment of inertia of a triangular lamina, Math. Gazette 39 (1955), 145.
[5]: R.A. Horn and C.A. Johnson, Matrix Analysis, Cambridge U.P., New York, 1985.
[6]: D. Kalman, The most marvelous theorem in mathematics, Journal of Online Mathematics and its Applications, available at
http://www.JOMA.org (for journal),
http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeID=1663 (direct to article).
[7]: D. Kalman, An elementary proof of Marden's Theorem, Amer. Math. Monthly 115 (Apr 2008), 330-338.
[8]: C. W. Kilmister, Some further remarks on moments of inertia, Math. Gazette, 44 (1960), 224-225.
[9]: D. Minda and S. Phelps, Triangles, Ellipses and Cubic Polynomials Amer. Math. Monthly (Oct 2008)
[10]: E.J. Routh, The Elementary Part of a Treatise on the Dynamics of a System of Rigid Particles, MacMillan and Co.: London, 1891.
[11]: A. Talbot, Equimomental systems, Math. Gazette 36 (1952), 95-110.

Grant Keady,
School of Mathematics and Statistics, University of Western Australia
email: keady@maths.uwa.edu.au

P. Scales,
Balanced Engineering Advice, Woodlands 6018, Australia
email: peterscales@perthpcug.org.au

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On 4 Jun 2008, 16:29.