Bounds for the CRDT conformal mapping algorithm (Q977103)

The Schwarz-Christoffel integral presents a conformal map from the unit disk \(\mathbb D\) onto the interior \(\Omega\) of a polygon \(P\). It contains the unknown parameters which are the preimages of the vertices of \(P\). Driscoll and Vavasis introduced the CRDT algorithm (Cross Ratios and Delaunay Triangulations) to compute the Schwarz-Christoffel parameters and formulated a number of conjectures to qualify the convergence. The author proves one of these conjectures and answers some of other questions. For \(\mathbf w=\{w_1,\dots,w_n\}\) and \(\mathbf z=\{z_1,\dots,z_n\}\), the distance between them in a quasiconformal sense is defined as \[ d_{QC}(\mathbf w,\mathbf z)=\inf\Big\{\log K:\exists \text{\(K\)-quasiconformal}\; h:\mathbb D\to\mathbb D\; \text{such that}\; h(\mathbf z)=\mathbf w\Big\}. \] Theorem 1: There is a \(C<\infty\) independent of \(P\) so that the initial guess \(\mathbf w\) of the CRDT algorithm satisfies \(d_{QC}(\mathbf w,\mathbf z)\leq C\), where \(\mathbf z\) are the true conformal prevertices. Theorem 2: Let \(\Gamma=\{\gamma_j\}\) denote the diagonals of the Delaunay triangulation of \(\Omega\), and suppose that any two points \(z,w\in\partial\Omega\) can be joined in \(\partial\Omega\cup\Gamma\) by a path of length at most \(M|z-w|\). Then \(d_{QC}(\mathbf w,\mathbf z)\leq C(M)\). A crescent is a domain bounded by two circular arcs meeting at two distinct points. If one of these arcs is a line segment, they bound a ''flat crescent''. A region \(R\) is constructed by adding a flat crescent to the outside of each edge of \(P\). Driscoll and Vavasis showed that a polygon \(P'\) obtained by adding extra vertices to \(P\) has Delaunay triangulation of three kinds of triangles: interior, thin isosceles, well separated. Theorem 4: Let \(P'\) be a polygon bounding a region \(\Omega\) and suppose \(P'\) has a triangulation whose elements are each one of the three types described above. Let \(R\) be the surface constructed from \(P'\) as above. Then there is a quasiconformal map \(\Psi:\Omega\to R\) that fixes each vertex of \(P'\) and has QC constant bounded independent of \(n\) and \(P'\). Theorem 5: Suppose \(P\) is a planar polygon with vertices \(\mathbf v=\{v_1,\dots,v_n\}\), and \(R\) is the Riemann surface constructed from a Delaunay triangulation of \(P\) as above. Then there is a continuous mapping of \(\Phi: R\to\mathbb D\) so that \(\Phi(\mathbf v)=\mathbf w\), where \(\mathbf v\) and \(\mathbf w\) are the initial guesses of the CRDT algorithm, and \(\Phi\) is a quasi-isometry between the hyperbolic metrics on \(R\) and \(\mathbb D\) with constants that are independent of \(n\) and \(P\).