Determine if projection of 3D point onto plane is within a triangle In 3D, given three points $P_1$, $P_2$, and $P_3$ spanning a non-degenerate triangle $T$. How to determine if the projection of a point $P$ onto the plane of $T$ lies within $T$?

The question is a slight extension of the question given here: Check whether a point is within a 3D Triangle
There is an elegant solution to this given by W. Heidrich, Journal of Graphics, GPU, and Game Tools,Volume 10, Issue 3, 2005.
Let $\vec{u}=P_2-P_1$, $\vec{v}=P_3-P_1$, $\vec{n}=\vec{u}\times\vec{v}$,
$\vec{w}=P-P_1$. We then have directly the barycentric coordinates of the projection $P'$ of $P$ onto $T$ as 


*

*$\gamma=\left[(\vec{u}\times\vec{w})\cdot\vec{n}\right]/\vec{n}^2$

*$\beta=\left[(\vec{w}\times\vec{v})\cdot\vec{n}\right]/\vec{n}^2$

*$\alpha=1-\gamma-\beta$
The coordinates of the projected point is


*

*$P'=\alpha P_1+\beta P_2 +\gamma P_3$
The point $P'$ lies inside $T$ if


*

*$0\leq\alpha\leq 1$, 

*$0\leq\beta\leq 1$, and

*$0\leq\gamma\leq 1$.

