# Map a 3D point into its corresponding 2D coordinates on a plane

My problem is the following:
I have three 3D points, having coordinates $$P_0 = (x_{p0}, y_{p0}, z_{p0})$$, $$P_1 = (x_{p1}, y_{p1}, z_{p1})$$, $$P_2 = (x_{p2}, y_{p2}, z_{p2})$$. I know the 2D coordinates of (the image of) each of these 3D points onto a 2D plane, namely $$T_0 = (x_{t0}, y_{t0})$$, $$T1_0 = (x_{t1}, y_{t1})$$, $$T_2 = (x_{t2}, y_{t2})$$

Given a chosen point $$Q = (x_{q0}, y_{q0}, z_{q0})$$ lying onto the plane defined by the 3D points, I need to find its corresponding 2D coordinates.

### Possible solutions

I found suggestions regarding the use of the barycenter of the 3D triangle, but I still cannot figure out how to do it.
I also thought of calculating transformation matrix, but that seems like an overcomplicated solution.

• Are you trying to map it into the z=0 plan?
– Moti
Nov 1, 2020 at 4:16
• How the projection is accomplished?
– Moti
Nov 1, 2020 at 4:17
• @Moti I am not trying to map it into the z=0 plan. The points T are not the orthogonal projection of the points P. They are obtained by a transformation, but I don't know the parameters of this trasformation. In practice, I am trying to retrieve the color of a point in 3D space from its corresponding texture map, which is defined on a plane. Nov 1, 2020 at 4:25
• I would assume a parallel projection and than see where the point falls to see if there is a single solution or it depends on the direction of the projection.
– Moti
Nov 1, 2020 at 6:36
• The plan you defined by removing z values could be (x,y) plan to which you are projecting your point - so the interpretation below as answer is right.
– Moti
Nov 3, 2020 at 6:35

I'm not so sure what you mean by defining 2D coordinates from a 3D plane, but it seems like you could just remove the Z axis, so $$Q=(x_{q0},y_{q0})$$.