1. To grind, erode, or shave something until it is made smooth or reduced in size. 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 "down." Make sure you plane down the sides of the car to reduce air resistance as much as possible. Erosion has planed the great cliffs down to a fraction of the size they were millions of years ago.
2. To grind, erode, or shave something until it has been totally removed (from something), leaving a smooth surface behind. A noun or pronoun can be used between "plane" and "down." This is just a rough cut of the timber—I still need to plane the imperfections down. The harsh weather planed down the names of those buried in the cemetery from their headstones.
plane something down
to smooth something down with a plane; to remove some material from something with a plane. I will have to plane the door down before I hang it again. I planed down the edge of the door for you.