Quadrilateral meshes with provable angle bounds

In this paper, we present an algorithm that utilizes a quadtree data structure to construct a quadrilateral mesh for a simple polygonal region in which no newly created angle is smaller than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{18.43}}^{\circ} ({=}\hbox{arctan}(\frac{1}{3}))$$\end{document} or greater than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{171.86}}^{\circ} ({=}{{135}}^{\circ} + 2\hbox{arctan}(\frac{1}{3}))$$\end{document}. This is the first known result, to the best of our knowledge, on a direct quadrilateral mesh generation algorithm with a provable guarantee on the angles.