  A conformal flattening 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 flattened
domain, or else demand expensive nonlinear optimization. Boundary first
flattening (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 efficiently
constructed via the Cherrier formula together with a pair of Poincare-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 flattening. 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 first time how a surface can be conformally
flattened directly onto any given target shape.
