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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the circumcenter of mass is a center associated with a
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
which shares many of the properties of the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
. More generally, the circumcenter of mass may be defined for
simplicial polytope In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinit ...
s and also in the
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
and
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
geometries. In the special case when the polytope is a
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
or
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
, the circumcenter of mass has been called the "quasicircumcenter" and has been used to define an
Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, includ ...
of a quadrilateral. The circumcenter of mass allows us to define an Euler line for simplicial polytopes.


Definition in the plane

Let P be an oriented polygon (with vertices counted countercyclically) in the plane with vertices V_1,V_2,\ldots,V_n and let O be an arbitrary point not lying on the sides (or their
extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
). Consider the triangulation of P by the oriented triangles O V_i V_ (the index i is viewed modulo n). Associate with each of these triangles its circumcenter C_i with weight equal to its oriented area (positive if its sequence of vertices is countercyclical; negative otherwise). The circumcenter of mass of P is the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of these weighted circumcenters. The result is independent of the choice of point O.


Properties

In the special case when the polygon is cyclic, the circumcenter of mass coincides with the
circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
. The circumcenter of mass satisfies an analog of Archimedes' Lemma, which states that if a polygon is decomposed into two smaller polygons, then the circumcenter of mass of that polygon is a weighted sum of the circumcenters of mass of the two smaller polygons. As a consequence, any triangulation with nondegenerate triangles may be used to define the circumcenter of mass. For an
equilateral polygon In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon ...
, the circumcenter of mass and center of mass coincide. More generally, the circumcenter of mass and center of mass coincide for a simplicial polytope for which each face has the sum of squares of its edges a constant. The circumcenter of mass is invariant under the operation of "recutting" of polygons. and the discrete bicycle (Darboux) transformation; in other words, the image of a polygon under these operations has the same circumcenter of mass as the original polygon. The generalized Euler line makes other appearances in the theory of integrable systems. Let V_i=(x_i,y_i) be the vertices of P and let A denote its area. The circumcenter of mass CCM(P) of the polygon P is given by the formula : CCM(P)=\frac(\sum_^ -y_i y_^2+y_i^2 y_ +x_i^2 y_-x_^2 y_i, \sum_^ -x_ y_i^2+x_i y_^2+x_i x_^2-x_i^2 x_). The circumcenter of mass can be extended to smooth curves via a limiting procedure. This continuous limit coincides with the center of mass of the homogeneous
lamina Lamina may refer to: Science and technology * Planar lamina, a two-dimensional planar closed surface with mass and density, in mathematics * Laminar flow, (or streamline flow) occurs when a fluid flows in parallel layers, with no disruption betwee ...
bounded by the curve. Under natural assumptions, the centers of polygons which satisfy Archimedes' Lemma are precisely the points of its Euler line. In other words, the only "well-behaved" centers which satisfy Archimedes' Lemma are the affine combinations of the circumcenter of mass and center of mass.


Generalized Euler line

The circumcenter of mass allows an
Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, includ ...
to be defined for any polygon (and more generally, for a simplicial polytope). This generalized Euler line is defined as the affine span of the center of mass and circumcenter of mass of the polytope.


See also

*
Circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
*
Circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircl ...


References

{{reflist Polygons Geometric centers Polytopes Mass