Barycentric coordinates

Remarks

Example: in $\R^2$ take $p_0 = (0, 0)$, $p_1 = (1, 0)$ and $p_2 = (0, 1)$. The point $(x, y)$ has barycentric coordinates $(1 - x - y, x, y)$. Check: the weights add up to $1$, and $(1 - x - y)(0, 0) + x(1, 0) + y(0, 1) = (x, y)$. The centroid $(1/3, 1/3)$ gets the coordinates $(1/3, 1/3, 1/3)$.

Non-example: the points $(0, 0)$, $(1, 0)$, $(2, 0)$ of $\R^2$ are collinear, so they are not affinely independent. For them the coordinates are not unique, and the point $(0, 1)$ has none at all.

When $k$ is $\R$, a point lies in the closed simplex with vertices $p_0, \dots, p_n$ exactly when all its barycentric coordinates are nonnegative. Homogeneous barycentric coordinates drop the condition $\sum \lambda_i = 1$: any tuple with nonzero sum is accepted, and two tuples that differ by a nonzero factor give the same point.