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  "body": "\n## Summary\n\nBarycentric coordinates describe a point of an [[affine space|09XNAPTJQ7]] by weights attached to the vertices of a fixed reference simplex.\nThe weights add up to $1$, and the point is the [[affine combination|0KJH6SRFZK]] of the vertices with these weights.\nThey need no origin and no basis, so they are the natural coordinates of affine geometry.\nThey are widely used for triangles, in computer graphics, and in finite element methods.\n\n## Explanations\n\n### L1 human\n\nPut masses on the three corners of a triangle; their balance point is a point of the plane.\nThe masses, scaled so that they add up to $1$, are the barycentric coordinates of that point.\nThe centroid gets the equal weights $(1/3, 1/3, 1/3)$, and each corner gets weight $1$ on itself and $0$ on the others.\nA point lies inside the triangle exactly when all three weights are positive.\nNegative weights are allowed and give the points outside the triangle.\n",
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