Project 3d lines onto plane

Question

I'm trying to wrap my head around coordinate projection, and can't seem to solve the problem I am having. Here is an illustration of it:

The cubes represent 3d space, and the 2 red lines represent the positions of 3 points (which could be n points). I'm then trying to project these lines onto the green plane shown on the second cube.

My code so far for this is the following:

private List<Vector3> ProjectPoints(List<Vector3> points, Vector3 axis1, Vector3 axis2)
{
    var projectedPoints = new List<Vector3>();

    // first point is already on plane
    projectedPoints.Add(points[0]);

    // blue line
    axis1.Normalize();
    // yellow line
    axis2.Normalize();

    var planeNormal = Vector3.Cross(axis1, axis2);

    float totalRotationAngleSoFar = 0;

    // project remaining points
    for (int i = 1; i < points.Count; i++)
    {
        // get point relative to origin (prev point)
        Vector3 pointRelativeToOrigin = points[i] - points[i - 1];
        Vector3 pointDir = Vector3.Normalize(pointRelativeToOrigin);

        // rotate point by all the angles from previous points, this is so if we had
        // a curve where the points went in on themselve, this rotation would see the
        // curve 'unfolding' into a straight line on the plane
        Vector3 rotationAxis = Vector3.Cross(pointDir, planeNormal);
        Vector3 rotatedPoint = Vector3.Transform(pointRelativeToOrigin, Matrix.CreateFromAxisAngle(rotationAxis, totalRotationAngleSoFar));

        // get rotation axis and min angle between point and plane.
        float minAngleBetweenPointAndPlane = TODO_GetAngleBetweenPointAndPlane();

        // add to total rotation for the next point to use (TODO minus or plus?)
        totalRotationAngleSoFar -= minAngleBetweenPointAndPlane;

        // rotate point in line with plane
        Vector3 projectedPoint = Vector3.Transform(rotatedPoint, Matrix.CreateFromAxisAngle(rotationAxis, minAngleBetweenPointAndPlane));
        projectedPoint += projectedPoints.Last();
        projectedPoints.Add(projectedPoint);
    }

    return projectedPoints;
}

I 'think' this is the right approach, though not 100% sure. Also I've no idea how to calculate TODO_GetAngleBetweenPointAndPlane() with the data I have. I expect you can use the dot product, but I'm not sure what data I would plug in.

Answer 1

I figured it out. Answer to my question was this:

// get rotation axis and min angle between point and plane.
var dot = Vector3.Dot(planeNormal, rotatedPointDir);
float angle = (float)Math.Acos(dot);
float minAngleBetweenPointAndPlane;
if (angle >= MathHelper.PiOver2)
    minAngleBetweenPointAndPlane = angle - MathHelper.PiOver2;
else
    minAngleBetweenPointAndPlane = MathHelper.PiOver2 - angle;

As well as storing the rotation matrices (instead of just angle), and also using the signed angle when using CreateFromAxisAngle.