Linear precision for parametric patches

We study linear precision for multi-sided parametric patches of any dimension, showing that every proper parametric patch has a unique reparametrization which has linear precision and giving a geometric criterion for when this reparametrization is rational. For toric patches, this geometric criterion is equivalent to a certain toric differential defining a birational map. While the reparametrization of a general toric patch having linear precision is not necesssarily a rational function, we show that it is computed by iterative proportional fitting, a numerical algorithm from statistics.