conformal parameterization
To paint a texture onto a 3D model, or to lay a curved surface flat without crumpling its detail, you need a map from the surface to a flat plane. A conformal parameterization is such a flattening that preserves angles: it may stretch the surface bigger here and smaller there, but it never shears — tiny circles on the surface map to tiny circles in the plane, never to ellipses. Mapmakers call this the Mercator idea: shapes are right locally even though Greenland's size is wildly distorted.
Formally, a parameterization is a map f from the surface (or a mesh patch) to the plane; it is conformal if its differential scales the metric by a positive function, df^T df = lambda * I for some lambda > 0 at each point, equivalently if it preserves angles between curves. On a triangle mesh you cannot usually achieve this exactly (a generic mesh has nonzero curvature concentrated at vertices), so 'discrete conformal' means satisfying a discrete angle condition. Several rigorous discretizations exist: least-squares conformal maps (LSCM) and the equivalent discrete conformal energy minimize the deviation from the Cauchy-Riemann equations and reduce to a sparse linear system built from the cotangent Laplacian; circle patterns and conformal equivalence of triangle meshes (Springborn-Schroder-Pinkall) preserve a discrete notion of length cross-ratios and lead to a convex variational problem solving for per-vertex scale factors. In each, the cotangent Laplacian or its complex cousin is the core operator.
Conformal parameterizations matter because angle preservation keeps textures and remeshing free of shear distortion, and because the underlying theory connects to deep mathematics — the uniformization theorem says every surface carries a conformal structure equivalent to a sphere, plane, or hyperbolic plane, and discrete conformal geometry is the computational shadow of that. The honest caveats: conformal maps generally cannot also preserve area (you trade one for the other — that is exactly why Mercator inflates the poles), a disk-type surface can be flattened but a closed surface of genus 0 needs a cut or a spherical/stereographic target, and higher-genus surfaces map conformally to the hyperbolic plane, not the Euclidean one. 'Conformal' fixes angles, never lengths.
Least-squares conformal mapping flattens a disk-like mesh patch by minimizing the discrete conformal energy E_C(u) = E_D(u) - A(u), where E_D is the Dirichlet energy (a cotangent-Laplacian quadratic form) and A is the signed area of the flattened image; pinning two vertices fixes the otherwise free translation/rotation/scale, and the result is a sparse least-squares solve whose output is shear-free up to the unavoidable area distortion.
Conformal flattening keeps angles right at the price of stretching area; pinning two points fixes the gauge.
Conformal does not mean isometric or area-preserving: angle preservation forces area distortion in general (that is the content of uniformization), so a conformal map that 'looks stretched' is behaving correctly, not failing. For a faithful unwrapping you must accept either angle error or area error — you cannot in general have neither.