Length ratios along a given line are invariant under affine transformations, and that's the case in Dan's problem.   --  Gene From: Warren D Smith <warren.wds@gmail.com> To: math-fun@mailman.xmission.com Sent: Friday, May 29, 2015 2:47 PM Subject: [math-fun] a few tiny comments To E.S: length ratios are NOT invariant under affine transformations of plane. Area ratios are. To S.W: by Lp norm I meant, not the distance formula in the plane, which is always the usual L2 formula, but rather the closeness of the grid G1 to the grid G2 reckoned in terms of all the N point-to-point distances, via Lp norm of N numbers. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)