To grind, erode, or shave something until it has been totally removed (from something), leaving a smooth surface behind. Originally reserved for references to the carpenter's tool known as a "plane," the phrase is often applied to anything that has a smoothing or erosive force. A noun or pronoun can be used between "plane" and "away." This is just a rough cut of the timber—I still need to plane the imperfections away. The harsh weather planed away the names of those buried in the cemetery from their headstones.