Basic Examples

Find the barycentric coordinates for the point {0.3,0.4} for the coordinate system {{1,1},{-1,1},{0,-1}}:
pts={{1,1},{-1,1},{0,-1}};​​p={0.3,0.4};​​
[◼]
BarycentricCoordinates
[pts,p]
In[]:=
{0.5,0.2,0.3}
Out[]=
Check the finding :
{0.5,0.2,0.3}.pts
In[]:=
{0.3,0.4}
Out[]=

Scope

Exact input leads to exact output:
[◼]
BarycentricCoordinates
[{{-3},{5}},{5/2}]
In[]:=

5
16
,
11
16

Out[]=
BarycentricCoordinates works in 1D:
[◼]
BarycentricCoordinates
[{{-3},{5}},{2.5}]
In[]:=
{0.3125,0.6875}
Out[]=
BarycentricCoordinates
works also in higher dimensions:
[◼]
BarycentricCoordinates
[{{-3.98,-3.56,2.78,-1.},{2.68,-0.44,-2.68,-2.08},{-2.46,-1.6,-4.08,4.78},{-4.84,0.04,-1.78,-1.40},{1.18,-3.6,-0.36,-4.36}},{-0.517,-1.702`,-1.699`,-1.068`}]
In[]:=
{0.1,0.3,0.2,0.15,0.25}
Out[]=
BarycentricCoordinates
works also in very high dimensions :
n=100;​​pts=RandomReal[{-5,5},{n+1,n}];​​p=RandomReal[{-1,1},n];​​result=
[◼]
BarycentricCoordinates
[pts,p]
In[]:=
{0.42748,0.126266,0.0693615,0.126766,0.101695,0.185458,0.00107506,0.214196,-0.131343,0.153637,0.0699973,0.199845,0.059208,0.458284,-0.0560479,0.110389,-0.330699,0.0333281,0.150265,0.21491,-0.0426564,0.559691,-0.132699,-0.0388817,-0.145802,-0.0405748,0.0662651,0.0170699,-0.157111,0.0434903,-0.48829,0.395702,-0.14331,0.075018,0.109029,-0.0193125,-0.0948896,-0.44234,0.216683,0.097782,-0.141793,-0.237312,0.0516467,0.374174,0.388064,0.105489,-0.356958,-0.0859644,0.185814,-0.155719,-0.00426896,-0.125776,-0.0587524,0.00876888,0.218041,-0.183955,-0.19474,-0.122702,0.537372,0.29969,0.125423,-0.200491,-0.247871,-0.161041,-0.192718,0.288294,-0.119117,-0.132887,-0.106769,-0.00444547,-0.113236,-0.113586,-0.182259,0.125785,-0.469811,0.0225466,-0.0106513,-0.126183,0.249656,-0.0310462,-0.194804,-0.095691,0.149648,0.125703,-0.104654,0.206386,0.112288,0.10052,-0.139793,-0.346926,0.0497164,0.0033637,0.103437,0.0591215,-0.113904,0.0607793,-0.0393336,0.235726,-0.233703,-0.100827,0.043303}
Out[]=
Check that the barycentric coordinates add up to 1:
Total[result]
In[]:=
1.
Out[]=
Recreate the point p and subtract it from the original:
Chop[result.pts-p]
In[]:=
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
Out[]=

Properties and Relations

The barycentric coordinates add up to 1:
Total
[◼]
BarycentricCoordinates
[{{-3.98,-3.56,2.78,-1.},{2.68,-0.44,-2.68,-2.08},{-2.46,-1.6,-4.08,4.78},{-4.84,0.04,-1.78,-1.40},{1.18,-3.6,-0.36,-4.36}},{-0.517,-1.702`,-1.699`,-1.068`}]
In[]:=
1.
Out[]=

Possible Issues

Dimensions of the points have to agree with each other:
[◼]
BarycentricCoordinates
[{{1,2,3},{3,4},{3,4,6}},{1,0.5,1}]
In[]:=
BarycentricCoordinates
::unequaldimensions
:Dimensions of the coordinate system and the point do not agree
$Failed
Out[]=
Dimensions of the points have to agree with each other:
[◼]
BarycentricCoordinates
[{{1,2,3},{3,4,6},{1,-1,-4}},{1,0.5,1}]
In[]:=
BarycentricCoordinates
::simplexmismatch
:For points of dimension 3​, the coordinate system should be composed of 4 points.
$Failed
Out[]=

Neat Examples

Interactively move the points of the coordinate system:
ManipulateGraphics{EdgeForm[Black],FaceForm[],Polygon[p[[;;3]]],Red,Text[#,p[[#]]+{0,0.25}]&/@Range[3]},PlotRange3,PlotLabel
[◼]
BarycentricCoordinates
[p[[;;3]],p[[4]]],{{p,{{1,1.4},{-1,1},{0,-2},{0.3,0.4}}},Locator}
In[]:=