Remarks on the Circumcenter of Mass

Given a homogeneous polygonal lamina $P$, one way to find its center of mass is as follows: triangulate $P$, assign to each triangle its centroid, taken with the weight equal to the area of the triangle, and find the center of mass of the resulting system of point masses. That the resulting point, $CM(P)$, does not depend on the triangulation, is a consequence of Archimedes’ Lemma: if an object is divided into smaller objects, then the center of mass of the compound object is the weighted average of the centers of mass of the parts, with the weights equal to the respective areas.

Replace, in the above construction, the centroids of the triangles by their circumcenters. The resulting weighted average is called the circumcenter of mass of the polygon $P$, denoted by $CCM(P)$. This point is well defined, that is, does not depend on the triangulation (assuming that degenerate triangles are avoided), see Fig. 1.

This construction is mentioned in the 19th century book ([Laisant1887]), where it is attributed to the Italian algebraic geometer G. Bellavitis. We learned about this reference from B. Grünbaum who, together with G. C. Shephard, studied this at about the same time, the circumcenter of mass was rediscovered by [Adler1993]; [Adler1995] as an integral of a discrete dynamical system called recutting of polygons.

The explicit formulas are as follows. Let the coordinates of the vertices of the polygon $P$, taken in the cyclic order, be $(x_{i},y_{i}),\ i=1,\ldots,n$. Then $CCM(P)=\frac{1}{4A(P)}\left(\sum_{i=0}^{n-1}y_{i}(x_{i-1}^{2}+y_{i-1}^{2}-x_{i+1}^{2} -y_{i+1}^{2}),\sum_{i=0}^{n-1}-x_{i}(x_{i-1}^{2}+y_{i-1}^{2}-x_{i+1}^{2}-y_{i+ 1}^{2})\right),$ where $A(P)$ is the signed area of $P$ (see [Tabachnikov and Tsukerman2014] for a proof). For comparison, $CM(P)=\frac{1}{6A(P)}\left(\sum_{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i} ),\sum_{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i})\right)$, a result of a straightforward calculation.

The construction of the circumcenter of mass extends to higher dimensions, and to the elliptic and hyperbolic geometries. We studied it in [Tabachnikov and Tsukerman2014] in relation with the so-called discrete bicycle transformation ([Tabachnikov and Tsukerman2013]). See also the paper by [Akopyan2014] .

The construction in ${\mathbb{R}}^{n}$ is similar. Given a simplicial polytope $P$, consider its non-degenerate triangulation. Assign the circumcenter $CC(\Delta_{i})$ to each simplex $\Delta_{i}$ of the triangulation, and take the center of mass of these points with weights equal to the oriented volumes of the respective simplices:

The result does not depend on the triangulation.

The explicit formula is as follows. Let $F=(V_{1},\ldots,V_{n})$ be a face of $P$, where $V_{i}$ are vectors in ${\mathbb{R}}^{n}$. Let $A(F)$ be the $n\times n$ matrix made of vectors $V_{i}$, and let $A_{i}(F)$ be obtained from $A(F)$ by replacing $i$th row with $(|V_{1}|^{2},\ldots,|V_{n}|^{2})$. Then the $i$th component of the circumcenter of mass is given by

In the above formulas, the signs of $\mathrm{Vol}(\Delta_{i})$ and the orders of vertices $V_{1},\ldots,V_{n}$ are chosen consistently: we consider the triangulation as a simplicial chain.

One can take affine combinations $tCM+(1-t)CCM,\ t\in{\mathbb{R}}$, resulting in a line, called the generalized Euler line of the polytope $P$ (for a triangle, the Euler line is the line through the centroid and the circumcenter; it passes through the orthocenter as well).

In this note we are interested in the uniqueness of this construction.

Suppose that to every non-degenerate simplex $\Delta\subset{\mathbb{R}}^{n}$ a ‘center’ $C(\Delta)\in{\mathbb{R}}^{n}$ is assigned so that the following assumptions hold:

1. 1.

   The map $\Delta\mapsto C(\Delta)$ commutes with similarities (both orientation-preserving and orientation-reversing);

2. 2.

   The map $\Delta\mapsto C(\Delta)$ is invariant under the permutations of the vertices of the simplex $\Delta$;

3. 3.

   The map $\varphi:\Delta\mapsto\mathrm{Vol}(\Delta)C(\Delta)$ is polynomial in the coordinates of the vertices of the simplex $\Delta$;

Let $x_{1},\ldots,x_{n}$ be Cartesian coordinates in ${\mathbb{R}}^{n}$. Let $\Delta=(V_{0},\ldots,V_{n})$ be a simplex, and let $V_{j}=(x_{1}^{j},\ldots,x_{n}^{j}),\ j=0,\ldots,n$, be the coordinates of its vertices (where $j$ is a superscript, not an exponent). Let

a multiple of the oriented volume of $\Delta$, and

where Skew is skew-symmetrization over superscripts. Evidently, $X_{i,jk}=X_{i,kj}$, and the number of such polynomials equals $n^{2}(n+1)/2$. Both determinants, $V$ and $X_{i,jk}$, are skew-symmetric under permutations of the vertices of the simplex.

Consider the map $\varphi:\Delta\mapsto\mathrm{Vol}(\Delta)C(\Delta)$, and let $(y_{1},\ldots,y_{n})$ be its components. Our assumption 3 implies that each $y_{\ell},\ \ell=1,\ldots,n$, is a polynomial in the variables $x_{1}^{0},x_{2}^{0},\ldots,x_{n}^{n}$. The assumption 1, applied to scaling, implies that these polynomials are homogeneous of degree $n+1$, and the assumption 2 that they are skew-symmetric under permutations of the superscripts. Lemma 2.1 implies that

where the coefficients $A_{i,jk}^{\ell}$ satisfy $A_{i,jk}^{\ell}=A_{i,kj}^{\ell}$. We always assume that summation is over repeated indices.

We now show that Archimedes’ Lemma is automatically satisfied for any choice of coefficients $A_{i,jk}^{\ell}$. Let $\Delta=(V_{0},\ldots,V_{n})$ be a simplex and $O$ a point. Consider the simplices

Introduce infinitesimal parallel translations in $r$th direction and rotations in the $p,q$-plane:

Let $\sigma$ denote a transposition of coordinates. The next lemma describes the action of these transformations on the polynomials $X_{i,jk}$.

The covariance of the map $\Delta\mapsto C(\Delta)$ with respect to rigid motions is expressed by the next equations on the coefficients $A_{i,jk}^{\ell}$.

Now we need to solve the system of linear Eqs. (3) – (5) on the unknowns $A_{i,jk}^{\ell}$. We use (3) to reduce the number of variables.

Consider the following four cases. If $|\{i,j,k,\ell\}|=4$ then, applying an appropriate sequence of transpositions, we obtain: $A_{i,jk}^{\ell}=A^{1}_{2,34}=:t$. If $|\{i,j,k,\ell\}|=3$, then one has four sub-cases, and $A_{i,jk}^{\ell}$ is equal to

Likewise, if $|\{i,j,k,\ell\}|=2$, then one has five sub-cases, and $A_{i,jk}^{\ell}$ is equal to

Finally, if $|\{i,j,k,\ell\}|=1$, then $A_{i,jk}^{\ell}=A^{1}_{1,11}=:\nu$. Thus we have 11 unknowns.

Now the strategy is to consider particular cases of (5) and (4).

To start with, consider (1) with $\ell=q=b=c\neq p=a$. One obtains

or $2\psi+\phi=\nu.$ Likewise, (4) with $\ell=r$ yields $(n-1)\psi+\nu=1/2.$ It follows that

We pause to check against Example 2.2. For the center of mass,

and the rest of variables vanish; for the circumcenter of mass, $\phi=\nu=1/2,$ and the rest vanishes. In both cases, (6) holds.

To finish the Proof of Theorem 1, we need to show that all variables, except $\phi,\psi,\nu$, vanish. We proceed in a similar fashion: (1) with $\ell=q=a=b=c\neq p$ yields

(1) with $\ell=q=c\neq p=a=b$ yields $\gamma=\alpha$. Hence $\beta=-\alpha$. Next, (4) with $\ell\neq r$ yields

hence $s=0$.

Next, (5) with $\ell=q=c$, but distinct from pairwise distinct $p,a,b$, yields $t=0$. It remains to eliminate $u,v,w$ and $\alpha$. Toward this, (5) with $\ell=q=c\neq p=b$ and distinct from $a$ yields

Likewise, (5) with $\ell=q=c\neq p=a$ and distinct from $b$, yields

Next, (5) with $\ell=q=c=a$, but distinct from pairwise distinct $p,b$, yields

and (5) $\ell=q=c=b$, but distinct from pairwise distinct $p,a$, yields $2v+w=0.$ We have obtained four linear equations on $u,v,w,\alpha$, and the only solution of this system is zero. This completes the proof.

(i) Degenerate simplices can be safely ignored when calculating the center of mass: such a simplex has a finite centroid and zero volume, making no contribution to the total sum. Not so for the circumcenter of mass: although the volume of a nearly degenerate simplex tends to zero, its circumcenter may go to infinity, and the contribution to the sum (1) may be non-negligible. The map $\varphi:\Delta\mapsto\mathrm{Vol}(\Delta)\ CC(\Delta)$, being polynomial in the coordinates of the vertices, is continuous.

For example, consider an isosceles right triangle $ABC$. Its circumcenter is the midpoint $M$ of the hypothenuse $AC$. Consider the triangulation in Fig. 2 consisting of three triangles, one of which, $AMC$, is degenerate. If one ignored this triangle, then, by Archimedes’ Lemma, the circumcenter of mass of $\triangle ABC$ would be the midpoint of the segment connecting the midpoints of the hypothenuses $AB$ and $BC$ of the triangles $ABM$ and $BCM$. The latter point is the circumcenter of the quadrilateral $ABCM$, not the triangle $ABC$.

(ii) One may wish to extend the notion of the circumcenter of mass to more general sets. For example, let $\gamma(t)$ be a parameterized smooth curve, star-shaped with respect to point $O$, see Fig. 2. It is natural to define the circumcenter of mass by continuity as

where $C(t)$ denotes the limiting $\varepsilon\to 0$ position of the vector from $O$ to the circumcenter of the infinitesimal triangle $O\gamma(t)\gamma(t+\varepsilon)$, and $dA$ is the area of this infinitesimal triangle. However, this does not give anything new: the integral (7) is the center of mass of the lamina bounded by the curve ([Tabachnikov and Tsukerman2014]).

(iii) Although the rational map $\Delta\mapsto CC(\Delta)$ is discontinuous, the polynomial map $\varphi:P\mapsto\mathrm{Vol}(P)\ CCM(P)$, defined on simplicial polytopes in ${\mathbb{R}}^{n}$, is continuous and is a valuation:

This valuation is isometry covariant; see, e.g., [Schneider2014] for the theory of valuations.

Continuity is with respect to the following topology in the space of polytopes in ${\mathbb{R}}^{n}$: if the vertices of a polytope $P$ are $V_{1},\ldots,V_{k}$, then we view $P$ as a point in ${\mathbb{R}}^{n}\times\ldots\times{\mathbb{R}}^{n}$ ($k$ times), and the topology is that of ${\mathbb{R}}^{nk}$.

(iv) We finish with open problems. Consider the same topology on the space of polytopes as in (iii).

As we mentioned, the construction of the circumcenter of mass extends to the spherical and hyperbolic geometries ([Akopyan2014]; [Tabachnikov and Tsukerman2014]). It is interesting to find an axiomatic description of the centers for simplicial polygons and polytopes, discussed in this note, in the three geometries of constant curvature (for the center of mass, see [Galperin1993]). These centers should be isometry covariant and satisfy some additivity condition (Archimedes’ Lemma or valuation-like). Such a description should include the valuations from Problem 4.1. At the moment of writing, we do not know such an axiomatic description.