  Seamless global parametrization of surfaces is a key operation in geometry
processing, e.g. for high-quality quad mesh generation. A common approach is to
prescribe the parametric domain structure, in particular the locations of
parametrization singularities (cones), and solve a non-convex optimization
problem minimizing a distortion measure, with local injectivity imposed through
either constraints or barrier terms. In both cases, an initial valid
parametrization is essential to serve as feasible starting point for obtaining
an optimized solution. While convexified versions of the constraints eliminate
this initialization requirement, they narrow the range of solutions, causing
some problem instances that actually do have a solution to become infeasible.
We demonstrate that for arbitrary given sets of topologically admissible
parametric cones with prescribed curvature, a global seamless parametrization
always exists (with the exception of one well-known case). Importantly, our
proof is constructive and directly leads to a general algorithm for computing
such parametrizations. Most distinctively, this algorithm is bootstrapped with
a convex optimization problem (solving for a conformal map), in tandem with a
simple linear equation system (determining a seamless modification of this
map). This initial map can then serve as valid starting point and be optimized
with respect to application specific distortion measures using existing
injectivity preserving methods.
