Map triangle and line from 3d to 2d

Question

I have a triangle in 3d and a line, they are lie in one plane. How can I map 5 points from 3d to 2d?

I can convert triangle to 2d, but can't do the same with line.

Vector2[] ConvertTo2d(Vector3 a, Vector3 b, Vector3 c)
{
    var aN = new Vector2(0, 0);
    var bN = new Vector2(0, 0);
    bN.x = Mathf.Sqrt(Mathf.Pow(b.x - a.x, 2f) + Mathf.Pow(b.y - a.y, 2f) + Mathf.Pow(b.z - a.z, 2f));
    var cN = new Vector2(0, 0);
    cN.x = ((b.x - a.x) * (c.x - a.x) + 
            (b.y - a.y) * (c.y - a.y) + 
            (b.z - a.z) * (c.z - a.z)) / bN.x;
    cN.y = Mathf.Sqrt(Mathf.Pow(c.x - a.x, 2f) + Mathf.Pow(c.y - a.y, 2f) + Mathf.Pow(c.z - a.z, 2f) - Mathf.Pow(cN.x, 2f));

    return new[] {aN, bN, cN};
}

What requirements do you have for the mapping? I'm guessing, it should be isometric. Anything else? There are still three degrees of freedom (rotation and translation in the 2D plane) — Nico Schertler, Apr 09 '19 at 01:58
I need map it in that way, that one of 3 coordinates of every point was equal to each other. For example Z coordinate. In that way I can use X and Y coordinates to get 2d points — Konstantin Saietskyi, Apr 09 '19 at 06:27
see the duplicate [How to create 2d plot of arbitrary, coplanar 3d curve](https://stackoverflow.com/a/44559920/2521214) use the equations with origin ... — Spektre, Apr 09 '19 at 08:18

Konstantin Saietskyi · Accepted Answer · 2019-04-09T13:54:53.843

Converted it to a code and it works!

var points = new Vector3[]
{
    triangleA,
    triangleB,
    triangleC,
    lineStart,
    lineEnd
};

var pointsIn2D = To2D(points);

Vector2[] To2D(Vector3[] points)
{
    var A = points[0];
    var B = points[1];
    var C = points[2];

    var U = B - A;
    var V = C - A;
    U /= U.magnitude;
    V /= U.magnitude;

    var W = Vector3.Cross(U, V);
    U = Vector3.Cross(V, W);

    Vector2[] pNew = new Vector2[points.Length];
    for (int i = 0; i < points.Length; i++)
    {
        var P = points[i];
        var xNew = Vector3.Dot(U, P);
        var yNew = Vector3.Dot(V, P);
        pNew[i] = new Vector2(xNew, yNew);
    }
    return pNew;
}

Map triangle and line from 3d to 2d

1 Answers1