Mathematics of surfaces XII

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This work explores various advanced topics in the topology and geometry of algebraic curves and surfaces. It begins with regularity criteria for algebraic structures and progresses to quadrangle surface tiling through contouring techniques. The discussion includes surfaces with piecewise linear support functions over spherical triangulations and examines developable surfaces characterized by uniformly negative internal angle deficits. The text delves into rational maximal parametrizations of Dupin cyclides and the computation of topology in arrangements of quartics. Further, it addresses non-uniform B-spline subdivision methods and scattered data fitting on surfaces via projected Powell-Sabin splines. Implicit boundary control for vector field-based shape deformations is presented alongside tuning subdivision algorithms through constrained energy optimization. The description of surfaces using parallel coordinates and linked planar regions is also covered. The theory and applications of discrete surface Ricci flow are examined, as well as the generation of guided C² spline surfaces with V-shaped tessellation. Additional topics include curvature estimation over smooth polygonal meshes, segmenting periodic reliefs on triangle meshes, and the estimation of end curvatures from planar point data. The work concludes with discussions on discrete surfaces in isotropic geometry, curvature-based surface regeneration, and methods for g