Lscm

Least Squares Conformal Maps (LSCM) is a method for mapping a triangular mesh in three dimensions onto a two-dimensional parameter domain in a way that minimizes angular distortion. It is widely used in texture mapping and mesh parameterization because it aims to preserve local angles (conformality) rather than areas.

The method formulates a linear least-squares problem by discretizing the Cauchy–Riemann equations. For each vertex i, a 2D coordinate z_i = (u_i, v_i) is sought. The conformality condition is enforced across neighboring triangles, leading to a sparse linear system in the unknowns {u_i, v_i} that encodes how the local triangles should rotate and scale to fit into the plane.

To fix the mapping up to a similarity transformation, boundary constraints are imposed by mapping the boundary vertices to a prescribed polygon, commonly a unit square or circle. This provides the necessary degrees of freedom to obtain a unique solution and yields a well-posed system.

Computationally, the resulting system is large, sparse, and linear, and can be solved efficiently with standard sparse linear solvers. The method is fast and robust for typical meshes, making it attractive for interactive or large-scale parameterization tasks.

Applications include generating texture coordinates (UV mapping), preprocessing for remeshing or morphing, and serving as a building block in more advanced parameterization pipelines. LSCM tends to produce low angular distortion, but areas may be stretched or compressed, with distortion often concentrating near the boundary depending on boundary conditions. It is one of several conformal or near-conformal parameterization techniques used in computer graphics, alongside harmonic maps and ARAP-based methods.