If your motion is correct, your putter head will work in a very specific manner. There has been a long discussion about the straight back path versus the slightly inside path. Take a look at the following illustration.
Do not confuse the target line with the swing path line. The putter head always stays square to you not to the ball. Because you are the centre of the circle, the correct putter path works as an arc away from the ball. However, the two lines match up close to the ball so that the face is square to the ball and you at impact
