Finding texture coordinates for plane

Question

I am creating a 3D ray tracer and want to add textured planes.

My planes are stored using a position vector and a normal vector.

I want to use a square texture and map it repetitively onto the plane.

How do I figure out a transformation from points (3D vectors) on the plane and points (2D vectors) on a texture? Once I know how to do this transformation, I don't think making it repetitive will be hard, as we can just apply modulo arithmetic.

I thought I could make a basis on the plane by calculating a vector B1 perpendicular to the normal, and then another one, B2, perpendicular to the normal and B1. But once I have these two vectors, I still don't know how to express the point (which is the argument for the function below) in this basis.

Vector3 position, normal;
Bitmap texture;

public override Vector3 GetColor(Vector3 point) {
    int u = ? 
    int v = ? 

    return texture.GetPixel(u, v);
}

Answer 1

To go from a point in worldspace, p = (x, y, z) and on the plane to texture coordinates u, v, you must first create a basis of 2 vectors, e1, e2 of the plane.

Given that N is the plane's normalized normal vector, we first create a vector perpendicular to this normal and some random other vector not parallel to the normal:

e1 = Vector3.Normalize(Vector3.Cross(N, new Vector3(1, 0, 0)));

//If normal and (1,0,0) are parallel, change e1
if (e1 == new Vector3(0, 0, 0)) {
    e1 = Vector3.Normalize(Vector3.Cross(N, new Vector3(0, 0, 1)));
}

And then we create the second basis vector by taking the cross product of the first basis vector and the normal, so that it is perpendicular to the normal, not parallel to the other basis vector, and thus in the plane:

e2 = Vector3.Normalize(Vector3.Cross(N, e1));

Then we can get the texture coordinates u, v by taking the dot product of the point in worldspace, p, with e1 and e2 respectively:

u = Vector3.Dot(e1, p);
v = Vector3.Dot(e2, p);

You might still have to do some scaling by texture sizes.

Source: UV World mapping in shader with Unity. and @DMGregory