The following graphics are composite of a chess piece's journey, gathered from over 2 million games from MillionBase. All of the moves are gathered together, and each faint line represents 500 moves from the dataset (rounding up).
Each arc is a very faint quadratic bezier curve, with the start and end point being the two squares on the board. The middle point is determined by a series of equations (basically the midpoint offset in both directions to give the arc both directionality and height).
As a hobby chess player, this is quite interesting to me and beautifully represented. From what I can tell, the more pronounced lines seem to be common opening moves for each of the pieces. The following is the visualization of the white knight. To view all the chess pieces, visit the Imgur album here or follow the creator _Tungs_ on twitter.