# Algebraic Representations for Volumetric Frame Fields

## TL;DR

This paper introduces a mathematically rigorous framework for representing and optimizing 3D volumetric frame fields, enabling more accurate and stable meshing in three-dimensional modeling.

## Contribution

It develops a differential and algebraic geometric understanding of octahedral frame spaces and proposes geometry-aware optimization tools, including semidefinite relaxations, for volumetric frame fields.

## Key findings

- Efficient algorithms produce high-quality, stable 3D frame fields.
- Mathematically sound description of octahedral and odeco frames.
- Generalization to anisotropic frames with independent axis scaling.

## Abstract

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.