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''of a set of points in Euclidean space''
 
''of a set of points in Euclidean space''
  
Consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c1101401.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c1101402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c1101403.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c1101404.png" />, with positive weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c1101405.png" /> attached to them. Then the centroid of this system is the point
+
Consider $  m $
 +
points $  a _ {i} $,  
 +
$  i = 1 \dots m $,  
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in $  \mathbf R  ^ {n} $,  
 +
with positive weights $  t _ {i} $
 +
attached to them. Then the centroid of this system is the point
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c1101406.png" /></td> </tr></table>
+
$$
 +
c = {
 +
\frac{\sum _ {i = 1 } ^ { m }  t _ {i} a _ {i} }{\sum _ {i = j } ^ { m }  t _ {i} }
 +
} .
 +
$$
  
This is also the centre of gravity, or barycentre, or centre of mass (in the sense of mechanics) of these mass points. The concept belongs to [[Affine geometry|affine geometry]] rather than to the vector space (Euclidean space) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c1101407.png" />, in the sense that the vector
+
This is also the centre of gravity, or barycentre, or centre of mass (in the sense of mechanics) of these mass points. The concept belongs to [[Affine geometry|affine geometry]] rather than to the vector space (Euclidean space) $  \mathbf R  ^ {n} $,  
 +
in the sense that the vector
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c1101408.png" /></td> </tr></table>
+
$$
 +
x + \left ( \sum _ {i = 1 } ^ { m }  t _ {i} \right ) ^ {- 1 } \sum _ {i = 1 } ^ { m }  t _ {i} ( a _ {i} - x )
 +
$$
  
is obviously independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c1101409.png" />.
+
is obviously independent of $  x $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014011.png" />, the term equi-barycentre is sometimes used. The concept can be extended to the case of possibly negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014012.png" /> (electric charges), see [[#References|[a1]]].
+
If $  t _ {i} = 1 $,  
 +
$  i = 1 \dots m $,  
 +
the term equi-barycentre is sometimes used. The concept can be extended to the case of possibly negative $  t _ {i} $(
 +
electric charges), see [[#References|[a1]]].
  
A related concept is that of the centre of a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014013.png" /> with non-empty interior in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014014.png" />, which is defined as:
+
A related concept is that of the centre of a compact set $  K $
 +
with non-empty interior in $  \mathbf R  ^ {n} $,  
 +
which is defined as:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014015.png" /></td> </tr></table>
+
$$
 +
c = \mu ( K ) ^ {- 1 } \int\limits _ { K } x  {d \mu } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014016.png" /> is [[Lebesgue measure|Lebesgue measure]]. This is the centre of mass, in the sense of mechanics, of the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014017.png" /> with uniform mass density. Here, a weight function (mass distribution) could be introduced.
+
where $  \mu $
 +
is [[Lebesgue measure|Lebesgue measure]]. This is the centre of mass, in the sense of mechanics, of the body $  K $
 +
with uniform mass density. Here, a weight function (mass distribution) could be introduced.
  
 
For a triangle, the centroid of the three corners, each with weight unity, is the intersection point of the three medians of the triangle (see [[Plane trigonometry|Plane trigonometry]]; [[Median (of a triangle)|Median (of a triangle)]]). This is also the centre of the convex hull of the three corners (Archimedes' theorem). This is no longer necessarily true for four points in the plane.
 
For a triangle, the centroid of the three corners, each with weight unity, is the intersection point of the three medians of the triangle (see [[Plane trigonometry|Plane trigonometry]]; [[Median (of a triangle)|Median (of a triangle)]]). This is also the centre of the convex hull of the three corners (Archimedes' theorem). This is no longer necessarily true for four points in the plane.
  
The centroid of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014018.png" /> with weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014020.png" />, minimizes the weighted sum of the squared distances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110140/c11014021.png" />.
+
The centroid of $  a _ {1} \dots a _ {m} $
 +
with weights $  t _ {i} $,  
 +
$  i = 1 \dots m $,  
 +
minimizes the weighted sum of the squared distances $  \sum _ {i = 1 }  ^ {m} t _ {i} \| {x - a _ {i} } \|  ^ {2} $.
  
 
See also [[Chebyshev centre|Chebyshev centre]] of a bounded set.
 
See also [[Chebyshev centre|Chebyshev centre]] of a bounded set.

Latest revision as of 16:43, 4 June 2020


of a set of points in Euclidean space

Consider $ m $ points $ a _ {i} $, $ i = 1 \dots m $, in $ \mathbf R ^ {n} $, with positive weights $ t _ {i} $ attached to them. Then the centroid of this system is the point

$$ c = { \frac{\sum _ {i = 1 } ^ { m } t _ {i} a _ {i} }{\sum _ {i = j } ^ { m } t _ {i} } } . $$

This is also the centre of gravity, or barycentre, or centre of mass (in the sense of mechanics) of these mass points. The concept belongs to affine geometry rather than to the vector space (Euclidean space) $ \mathbf R ^ {n} $, in the sense that the vector

$$ x + \left ( \sum _ {i = 1 } ^ { m } t _ {i} \right ) ^ {- 1 } \sum _ {i = 1 } ^ { m } t _ {i} ( a _ {i} - x ) $$

is obviously independent of $ x $.

If $ t _ {i} = 1 $, $ i = 1 \dots m $, the term equi-barycentre is sometimes used. The concept can be extended to the case of possibly negative $ t _ {i} $( electric charges), see [a1].

A related concept is that of the centre of a compact set $ K $ with non-empty interior in $ \mathbf R ^ {n} $, which is defined as:

$$ c = \mu ( K ) ^ {- 1 } \int\limits _ { K } x {d \mu } , $$

where $ \mu $ is Lebesgue measure. This is the centre of mass, in the sense of mechanics, of the body $ K $ with uniform mass density. Here, a weight function (mass distribution) could be introduced.

For a triangle, the centroid of the three corners, each with weight unity, is the intersection point of the three medians of the triangle (see Plane trigonometry; Median (of a triangle)). This is also the centre of the convex hull of the three corners (Archimedes' theorem). This is no longer necessarily true for four points in the plane.

The centroid of $ a _ {1} \dots a _ {m} $ with weights $ t _ {i} $, $ i = 1 \dots m $, minimizes the weighted sum of the squared distances $ \sum _ {i = 1 } ^ {m} t _ {i} \| {x - a _ {i} } \| ^ {2} $.

See also Chebyshev centre of a bounded set.

References

[a1] M. Berger, "Geometry" , I , Springer (1987) (Translated from French)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) (Edition: Second)
How to Cite This Entry:
Centroid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centroid&oldid=46299
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article