Constructing diffeomorphisms between simply connected plane domains

. Consider a simply connected domain Ω ⊂ R 2 with boundary ∂ Ω that is given by a smooth function ϕ : [ a,b ] (cid:55)→ R 2 . Our goal is to calculate a diffeomorphism Φ : B 1 (0) (cid:55)→ Ω , B 1 (0) the open unit disk in R 2 . We present two different methods where both methods are able to handle boundaries ∂ Ω that are not star-shaped. The first method is based on an optimization algorithm that optimizes the curvature of the boundary, and the second method is based on the physical principle of minimizing a potential energy. Both methods construct first a homotopy between the boundary ∂ B 1 (0) and ∂ Ω and then extend the boundary homotopy to the inside of the domains. Numerical examples show that the method is applicable to a wide variety of domains Ω .