The natural triangular coordinate system is defined by a horizontal $\xi$ or $r$ axis and a vertical $\eta$ or $s$ axis. Both axes range from 0 to 1.
Image source: Yunhua, L. & Eriksson, A. "An alternative assumed strain method." Computer Methods in Applied Mechanics and Engineering, Vol 178, 1999, Pages 23-37.
The coordinates $\xi$ (or $r$) and $\eta$ (or $s$) can be computed from triangular area coordinates.
Area coordinates for a triangle are defined by the image shown below:
The areas of each triangle are given below:
$A=\frac{1}{2} \left[(x_2 y_3 - x_3 y_2)+(x_3 y_1-x_1 y_3)+(x_1 y_2-x_2 y_1)\right]=$ Total element area
$A_1=\frac{1}{2} \left[(x_{3} y_{0}-x_{0} y_{3})+(x_{0} y_{2}-x_{2} y_{0})+(x_{2} y_{3}-x_{3} y_{2})\right]$
$A_2=\frac{1}{2} \left[(x_{1} y_{0}-x_{0} y_{1})+(x_{0} y_{3}-x_{3} y_{0})+(x_{3} y_{1}-x_{1} y_{3})\right]$
$A_3=\frac{1}{2} \left[(x_{2} y_{0}-x_{0} y_{2})+(x_{0} y_{1}-x_{1} y_{0})+(x_{1} y_{2}-x_{2} y_{1})\right]$
The triangular area coordinates are:
$\zeta_1=\frac{A_1}{A}$ $\zeta_2=\frac{A_2}{A}$ $\zeta_3=\frac{A_3}{A}$
The natural triangular coordinates are:
$\xi, r=\zeta_2$
$\eta, s=\zeta_3$
Given a natural coordinate, $(\xi, \eta)$, the local coordinates $(x, y)$ can be recovered using the following:
$x=(x_1-x_3) \xi+(x_2-x_3) \eta+x_3$