  Field-guided parametrization methods have proven effective for quad meshing
of surfaces; these methods compute smooth cross fields to guide the meshing
process and then integrate the fields to construct a discrete mesh. A key
challenge in extending these methods to three dimensions, however, is
representation of field values. Whereas cross fields can be represented by
tangent vector fields that form a linear space, the 3D analog---an octahedral
frame field---takes values in a nonlinear manifold. In this work, we describe
the space of octahedral frames in the language of differential and algebraic
geometry. With this understanding, we develop geometry-aware tools for
optimization of octahedral fields, namely geodesic stepping and exact
projection via semidefinite relaxation. Our algebraic approach not only
provides an elegant and mathematically-sound description of the space of
octahedral frames but also suggests a generalization to frames whose three axes
scale independently, better capturing the singular behavior we expect to see in
volumetric frame fields. These new odeco frames, so-called as they are
represented by orthogonally decomposable tensors, also admit a semidefinite
program--based projection operator. Our description of the spaces of octahedral
and odeco frames suggests computing frame fields via manifold-based
optimization algorithms; we show that these algorithms efficiently produce
high-quality fields while maintaining stability and smoothness.
