Convert a polygone in space, to a plane

Question

I have a surface defined by a list of 3D points (X,Y,Z). (so all vectors are coplanars) I would like to convert that on a plane (X,Y). I am looking for a formula to convert that polygon using vectors, but until now only found a solution offering this :

• Calculate length of each segment.
• Calculate angles between each of them
• Then I can find the X,Y coordinates with that.

But I think there may exist a vectorial solution for that?

Thanks in advance

Answer 1

Find two vectors parallel to the plane by taking differences of points in the plane, such as $\vec v_1=(x_1,y_1,z_1)-(x_0,y_0,z_0),\;\vec v_2=(x_2,y_2,z_2)-(x_0,y_0,z_0)$, then use the Gram-Schmidt process on $\vec v_1,\vec v_2$ to get an orthonormal basis $\hat u_1,\hat u_2$ for the plane, and finally take dot products to get points $(x',y')$ in 2D: $x'=(x,y,z)\cdot\hat u_1,\;y'=(x,y,z)\cdot\hat u_2$.