Move Parent Along Childs Local Axis

I’m building a 3D puzzle in which the camera rotates around the puzzle on a central axis. I want to be able to move the pieces relevant to the camera direction only on X and Y.

The piece has a plane within it as a child. This plane is always facing the camera, this plane is what detects the Raycast. I want to be able to move the piece along the plane axis on X and Y.

Below as an image of the piece selected (I’m looking through the game camera), and as you can see, the axis is incorrect for the movement of X and Y relevant to the camera. The purple plane is the plane I mentioned above, it always faces the camera.

As you can see, when I select the plane, the axis is correct. So I need to somehow move the piece (plane’s parent), along this axis only on X and Y.

How can I achieve this? The raycast update is called like this on the puzzle piece:

//hitPoint is the location of the raycast on PLANE
function UpdatePiecePosition(hitPoint : Vector3){

	//right now - I'm just doing a general position
	//raycast plane is moved as well as its a child - this is fine
	transform.position = hitPoint;

	//update planes rotation to camera

Check the Transform.TransformDirection and Transform.InverseTransformDirection methods. You can get, say, the transform.right of the plane (which returns the right direction in local space):

Vector3 right = plane.transform.right;

… trasform that direction into a world space direction:

right = plane.transform.TransformDirection(right);

… and then transform that direction into the local space of the parent game object:

right = theParentObject.transform.InverseTransformDirection(right);

You can now move the object in that direction.