barycentric coordinate calculation correction

[ntoussaint]

Barycentric coordinate calculation

I seem to have found a problem in the barycentric coordinate calculation.

The original implementation was giving me incorrect coordinates. This seemed to not influence the "distance", and therefore might have been remained overlooked.

The correction is simple and is not adding computational time.

The calculation originates from this math-stackexchange comment

Let $\overrightarrow{u} = b - a$, $\overrightarrow{v} = c - a$, $\overrightarrow{w} = p - a$ and $\overrightarrow{n} = \overrightarrow{u} \times \overrightarrow{v}$. The barycentric coordinates of $p'$, the projection of $p$ onto the plane $(a,b,c)$ are therefore:

$$ \gamma = [(\overrightarrow{u} \times \overrightarrow{w} ) \cdot \overrightarrow{n} ] / | \overrightarrow{n} |^2 $$

$$ \beta = [(\overrightarrow{w} \times \overrightarrow{v} ) \cdot \overrightarrow{n} ] / | \overrightarrow{n} |^2 $$

$$ \alpha = 1 - \beta - \gamma$$

The coordinates of the projected point is

$$ p' = \alpha a+ \beta b + \gamma c $$