In this lecture we’ll attempt to translate several smooth characterizations of conformal maps into the discrete setting. We’ll also see how each of these attempts leads to a different class of computational algorithms, with different trade offs. Although conformal maps are often associated with angle preservation, we’ll see that preserving angles at the discrete level results in a theory that is far to rigid: i.e., it does not capture any of the interesting structure of smooth conformal geometry. Instead, we’ll see how less-obvious starting points (such as patterns of circles, or preservation of length cross ratios) lead to some deep and beautiful connections between the classic smooth picture, and the discrete, combinatorial picture.
