How can I describe a triangle or tetrahedron with barycentric coordinates?

Question

How would I, for example, describe a triangle given by the vertices $(2,1),(4,2),(2.5,2)$ with barycentric coordinates?

My guess would be $(a:b:1-a-b)$ where $0\leq a,b\leq 0.75$ and $a+b\leq 0.75$ since $A_\Delta=0.75$.

Is that true?

Answer 1

Here is how I would describe it -

If we consider masses $(\alpha, \beta, \gamma)$ placed at the vertices of the $\triangle ABC, A(2,1), B(4,2), C(2.5,2) \,$ respectively then all points in the triangle can be described as -

$(2 \alpha + 4 \beta + 2.5 \gamma, \alpha + 2 \beta + 2 \gamma)$

$0 \leq \alpha, \beta, \gamma \leq 1, \alpha + \beta + \gamma = 1$

or substituting $(\gamma = 1 - \alpha - \beta)$, we can represent all points in the triangle as $(2.5 - 0.5 \alpha + 1.5 \beta, - \alpha +2 ), 0 \leq \alpha, \beta \leq 1, \alpha + \beta \leq 1$.

Vice versa, we can find barycentric coordinates $(\alpha, \beta, 1 - \alpha - \beta)$ of any given point in the triangle using the above.

As you mentioned in your working, for any point $P$, the ratio of area can also be used to find its barycentric coordinates.

$\alpha = \frac{\triangle PBC}{\triangle ABC}, \beta = \frac{\triangle PCA}{\triangle ABC}, \gamma = \frac{\triangle PAB}{\triangle ABC} \,$ or $\, (1 - \alpha - \beta)$