Geometric convergence is tricky. Specifically, the issue here is that uniform convergence does not imply convergence in length or area or any other such measure

Geometric convergence is tricky. Specifically, the issue here is that uniform convergence does not imply convergence in length or area or any other such measures. (see figure 6 here: http://www.sci.utah.edu/~etiene/publications/verifiable-vis.pdf http://www.sci.utah.edu/~etiene/publications/verifiable-vis.... . disclaimer - I'm a co-author) Imagine a circle and its diameter. Now imagine two circles with half of the diameter, lined up so that the diameter lines align. Now split those two in four, etc. The circle becomes a snaking line whose total length doesn't change, and the snaking line converges uniformly to the line. Clearly, however, pi is not 1. What you need is convergence in position _and_ angle. A curve that converges in position and angle _does_ converge in length: the reference I know which shows this is reference [9] on the above-mentioned paper. (edit: in case you don't want to download the gigantic file --- yay for publishing in graphicsy places --- the reference is: K. Hildebrandt, K. Polthier, and M. Wardetzky. On the convergence of metric and geometric properties of polyhedral surfaces. Geometriae Dediacata, (123):89–112, 2006.)