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Thank you, Charles. I was intending to discuss the definition, with an interest in extending it a bit. I do see a possible need for expanded definitions [and a way to look up all pieces would be nice, even if all but impossible.] Planar pieces as used are just one step into area effect pieces - hook movers being another similar but different example, and the 2-step bent knightriders, especially [Jeremy's] with a ferz or wazir component, a third, shortrange version. Personally, I see a great need for some sort of organization of/within CVs. I'm happy there are people [crazy enough] um, willing to tackle it.   

I'd think planar pieces might be classified by how easy it is to block them, as a start. As pieces, they get more powerful the closer they are to their intended victim, an unusual phenomenon in chess, I suspect. This might be another sort of defining characteristic. I'd think they would be interesting pieces on very large boards, or greater than 2D boards. Which brings up the question of whether 'Planar' will be the stand-in [or the 'type specimen'] for planar, cubic, quatric, quintic... pieces. I'd say yes, at least for now. 

The more powerful versions of a planar piece would require more powerful versions of other pieces, or something such as a traffic cop piece, that slowed everything down, a la Roberto Lavieri's piece George mentioned recently, a somewhat less-than-Immobilizer. Or possibly some sort of reactive piece - Jeremy's Actualized Potential Pieces come to mind as a start for a piece that fires to a preset point or distance during the other player's turn, for example. Certainly, that piece would guard against the current planar pieces from 'behind' [but not 'directly' behind] the target piece. A 'battery' of these pieces might break up a major offensive in a larger [multimove, maybe] game. 

I guess the point of the previous paragraph is that once you've gotten the pieces to the board and used a bit, someone else will escalate. [Blue-sky fantasy desire: see all pieces cataloged in such a way that a designer who knew the system could look up newly-designed pieces to see if they are actually new, and what pieces are similar.] I'd think you'd want your system to be able to accomodate the pieces that will be showing up in the near future.