Editing Centroid (section)

== Properties ==
The geometric centroid of a [[convex set|convex]] object always lies in the object.  A non-convex object might have a centroid that is outside the figure itself. The centroid of a [[Annulus (mathematics)|ring]] or a [[bowl (vessel)|bowl]], for example, lies in the object's central void.

If the centroid is defined, it is a [[fixed points of isometry groups in Euclidean space|fixed point of all isometries]] in its [[symmetry group]]. In particular, the geometric centroid of an object lies in the intersection of all its [[hyperplane]]s of [[symmetry]].  The centroid of many figures ([[regular polygon]], [[regular polyhedron]], [[cylinder (geometry)|cylinder]], [[rectangle]], [[rhombus]], [[circle (geometry)|circle]], [[sphere (geometry)|sphere]], [[ellipse]], [[ellipsoid]], [[Lamé curve|superellipse]], [[superellipsoid]], etc.) can be determined by this principle alone.

In particular, the centroid of a [[parallelogram]] is the meeting point of its two [[diagonal]]s.  This is not true of other [[quadrilateral]]s.

For the same reason, the centroid of an object with [[translational symmetry]] is undefined (or lies outside the enclosing space), because a translation has no fixed point.