Transform points on a plane in $R^3 $ to the X-Y plane

Question

I have a group of points $p_1,p_2,..p_n$ where $p_n = (x_n, y_n, z_n)$ that are on a plane $ax + by + cz + d = 0$. I want to transform these points onto the X-Y plane (not project) so that $p_n^* = (x_n^*, y_n^*, 0)$ while preserving the distances between all the points.

Answer 1

Use the Gram-Schmidt process on $(p_2-p_1,p_3-p_1)$ to get $e_1$ and $e_2$, two orthonormal vectors. Then $(p_1,e_1,e_2)$ is a coordinate system.

Then you have $$x_n^*=(p_n-p_1)\cdot e_1$$ and $$y_n^*=(p_n-p_1)\cdot e_2$$