Difference between revisions of "Centroid"
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''of a set of points in Euclidean space'' | ''of a set of points in Euclidean space'' | ||
| − | Consider | + | 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|affine geometry]] rather than to the vector space (Euclidean space) | + | 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 | ||
| − | + | $$ | |
| + | x + \left ( \sum _ {i = 1 } ^ { m } t _ {i} \right ) ^ {- 1 } \sum _ {i = 1 } ^ { m } t _ {i} ( a _ {i} - x ) | ||
| + | $$ | ||
| − | is obviously independent of | + | is obviously independent of $ x $. |
| − | If | + | 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 | + | 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 | + | 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 | + | 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) |
Centroid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centroid&oldid=46299