= Subdivision surface
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A <subdivision surface> is a limit shape defined by repeated refinement of a <control net> using a specified <subdivision mask>. Each step changes the mesh connectivity and computes new <control points> from nearby old ones. When the scheme converges, progressively refined polygonal meshes approach the limit; the finite mesh at any chosen level is generally an approximation to that limit.
In <Loop subdivision>, each triangular face splits into four smaller triangles, and both old-vertex positions and new edge-vertex positions are weighted averages. These local rules allow a mesh with many different <degrees of vertices> to represent a smooth shape without constructing one global rectangular parameter grid. On a regular patch, a known <box spline> describes the limit. Near an <extraordinary subdivision vertex>, the <subdivision matrix> and <characteristic map of a subdivision surface> govern convergence and tangent behavior.
Rendering a refined mesh and querying the exact limit are different tasks. <Subdivision surface interrogation> evaluates a limit point, <derivatives> and a <normal vector> where they exist, and needs bounds that enclose the limit. Additional refinement makes an approximation finer, but alone does not prove a specific geometric error bound or guarantee <Gaussian curvature> exists at every extraordinary point.<Convex Triangular Subdivision Surfaces with Bounded Curvature by Charles Loop>[Introduction and section 2: recursive triangular refinement, regular valence six and extraordinary-vertex analysis]{ref}
// Target: analysis.bigb
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