Find angle between 2 points ignoring origin.forward in Unity

Question

Background: I am creating an AR treasure hunt app. It is simple, it has a locator that tells you where the treasure is relative to you. I have the camera being the origin and the treasure being an object in AR world.

Question: I would like to rotate my arrow according to where in space the treasure is at. but in 2d. Basically, I would ignore the relative forward plane that is camera.forward.

Example: If the camera rotation is default, the angle can be calculated as atan2(dy,dx). If the camera is looking straight down, the angle is atan2(dz,dx).

What I have tried:

Quaternion lookRot = Quaternion.LookRotation(target.transform.position - origin.transform.position);
Quaternion relativeRot = Quaternion.Inverse(origin.transform.rotation) * lookRot;

Relative rotation is correct in 3d space but I would like to convert that into 2d ignoring the camera.forward plane. So even if the treasure is in front or behind the camera, it should not change the angle.

Answer 1

Turns out there is a very simple way to get relative X and Y of the target.

Vector2 ExtractRelativeXY(Transform origin, Transform target) {
    // Get the absolute look rotation from origin to target.
    Quaternion lookRot = Quaternion.LookRotation(target.transform.position - origin.transform.position);
    // Create a relative look rotation with respect to origin's forward.
    Quaternion relativeRot = Quaternion.Inverse(origin.transform.rotation) * lookRot;
    // Obtain Matrix 4x4 from the rotation.
    Matrix4x4 m = Matrix4x4.Rotate(relativeRot);
    // Get the 3rd column (which is the forward vector of the rotation).
    Vector4 mForward = m.GetColumn(2);
    // Simply extract the x and y.
    return new Vector2(mForward.x, mForward.y);
  }

Once obtained x and y, turn it into angle using angle = atan2(y,x) as suggested by both MBo and Tom.

This works because of the matrix components of the quaternion can be demonstrated in multiple vectors. Better illustration is found here https://stackoverflow.com/a/26724912.