Boundary First Flattening

Abstract

A conformal attening maps a curved surface to the plane without distorting angles-such maps have become a fundamental building block for problems in geometry processing, numerical simulation, and computational design. Yet existing methods provide little direct control over the shape of the attened domain, or else demand expensive nonlinear optimization. Boundary rst attening (BFF) is a linear method for conformal parameterization which is faster than traditional linear methods, yet provides control and quality comparable to sophisticated nonlinear schemes. The key insight is that the boundary data for many conformal mapping problems can be e ciently constructed via the Cherrier formula together with a pair of Poincaré-Steklov operators; once the boundary is known, the map can be easily extended over the rest of the domain. Since computation demands only a single factorization of the real Laplace matrix, the amortized cost is about 50x less than any previously published technique for boundary-controlled conformal attening. As a result, BFF opens the door to real-time editing or fast optimization of high-resolution maps, with direct control over boundary length or angle. We show how this method can be used to construct maps with sharp corners, cone singularities, minimal area distortion, and uniformization over the unit disk; we also demonstrate for the rst time how a surface can be conformally attened directly onto any given target shape.

A Novel Algorithm for Volume-Preserving Parameterizations of 3-Manifolds

“…Additionally, for parameterizations of balancing distortions, some well-developed numerical methods, including the as-rigid-as-possible surface parameterization [51,72], the most isometric parametrization [41,20], the isometric distortion minimization [59], and boundary first flattening [61], have been proposed to reach a trade-off between minimizing the angle and area distortions.…”

A Subspace Method for Fast Locally Injective Harmonic Mapping

“…On the other hand, methods that use spatial coordinates result in a metric that is completely flat as long as the induced map is locally injective. The method of [SC17] tries to benefit from both worlds by first using the conformal factors as variables to induce a nearly flat metric, and then switches to the spatial variables in which conformality is discretized based on the Cauchy‐Riemann equations with a pair of harmonic conjugate functions. We note that while conformal maps possess elegant theory and many useful mathematical properties, they are somewhat more restricted than necessary, sometimes engendering large isometric (scale) distortion.…”

“…[AL15] uses the former result and generalizes the bijection such that it can be applied to genus 0 surfaces with either 3 or 4 cone singularities of a specific nature. As a last example, the recent method of [SC17] computes a discrete approximation to the unique conformal parametrization, given arbitrary prescription of cone singularities. It is more general than the former methods at the expense of not having strict injectivity guarantees (though it is empirically shown to be very robust).…”

Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills

“…Free boundary algorithms allow unrestricted boundary parameterizations, producing less distorted maps. Sawhney and Crane [11] presented an algorithm in which the mesh boundary is mapped to R 2 according to its shape. The parameterized boundary is then used as a constraint to produce a boundary-free parameterization.…”