Converting a 3D polygon into a 2D polygon

Questions & Answers
1

Suppose you have a set of coordinates on a 3D plane forming a polygon . I want to convert this 3d polygon into a 2d polygon with same lengths . How do I go about this? Are there any methods in unity that can help me do this

2

If your points are all on the same plane you just need to project them onto two unit vectors on that plane

To determine the plane normal you can simply calculate the cross product between two edges of your polygon. So just pick 3 points (A, B and C) and do

Vector3 normal = Vector3.Cross(B-A, C-A).normalized;

Next you want to define your first unit vector on that plane. If you don’t care about the rotation you can simply use one of the edges. Otherwise you may want to project one of the world axes onto the plane. You should make sure to not pick a direction that is too close to the plane normal

Vector3 u;
If (Mathf.Abs(Vector3.Dot(Vector3.forward, normal)) < 0.2f)
    u = Vector3.ProjectOnPlane(Vector3.right, normal);
else
    u = Vector3.ProjectOnPlane(Vector3.forward, normal);

To get the second unit vector just calculate the cross product between the first unit vector and the plane normal:

Vector3 v = Vector3.Cross(u, normal).normalized;

Now you can simply project each point of your polygon onto both unit vectors using the dot product. The two resulting values are your 2d coordinates of your points

Vector2 Project(Vector3 aPoint)
{
    return new Vector2(Vector3.Dot(aPoint, u), Vector3.Dot(aPoint, v));
}

Keep in mind that the origin will still be the same. So you may want to offset all points by your desired pivot position. If you have the pivot as a 3d point on your 3d plane you can project it the same way as you do with the other points. Just subtract the pivot from all points in the end and they will be relative to that point.

If you need more help you should be more specific about your issue. Note that if the 3d plane is aligned with one of the 3 worldspace planes things could be simplified by quite a bit. Though the solution i’ve posted here works for arbitrarily rotated planes in space. Though keep in mind that a plane in 3d can be viewed from both sides so the winding order of your polygon may be important. You may need to flip your plane normal

Note that if your polygon points are not in the same plane you will get some sort of distortion. If you’re looking for a way to unwrap a 3d mesh that’s a completely different topic and doesn’t have one generic solution.

3

@Bunny83 Thanks for that great and detailed answer! That is exactly what I was looking for.
Now I am stucking with the reverse transformation. I want to calculate points within the just transformed 2D plane and project these 2d points on my old 3d plane. I googled it but dont find a good approach. Do you have an idea?

4

This function will do that. You may want to set the reduced dimension to zero to avoid the floating point approximation to zero.

List<Vector3> alignVerticesOnPlane(List<Vector3> vertices)
     {
         List<Vector3> verticesOnPlane = new List<Vector3>();
 
         //1. Subtract vertices[0] from all points
         for (int i = 0; i < vertices.Count; i++)
             verticesOnPlane.Add(vertices *- vertices[0]);*

//2. Find the normalized plane normal
Vector3 A = verticesOnPlane[1];
Vector3 B = verticesOnPlane[2];
Vector3 n = (Vector3.Cross(A, B) / Vector3.Magnitude(Vector3.Cross(A, B)));

//3. Create a quaternion with the normal of the plane and the direction you want it to face
Quaternion rotation = Quaternion.FromToRotation(n, Vector3.forward); //change Vector3.forward to align with other planes

//4. Apply the quaternion rotation to all points
for (int i = 0; i < verticesOnPlane.Count; i++)
verticesOnPlane = rotation * verticesOnPlane*;*

return verticesOnPlane; // =)
}