# Controlling Meshes via Curvature: Spin Transformations for Pose-Invariant Shape Processing

## TL;DR

This paper introduces discrete spin transformations as a stable, curvature-based method for pose-invariant shape processing of surface meshes, enabling applications like surface fairing and mesh extrusion with minimal distortion.

## Contribution

It expands the discrete formulation of spin transformations, allowing controlled curvature manipulation for shape analysis and processing without remeshing, and demonstrates their effectiveness on complex biological structures.

## Key findings

- Stable shape manipulation across large deformations
- Effective surface fairing and mesh extrusion
- Controlled trade-off between area and angle distortions

## Abstract

We investigate discrete spin transformations, a geometric framework to manipulate surface meshes by controlling mean curvature. Applications include surface fairing -- flowing a mesh onto say, a reference sphere -- and mesh extrusion -- e.g., rebuilding a complex shape from a reference sphere and curvature specification. Because they operate in curvature space, these operations can be conducted very stably across large deformations with no need for remeshing. Spin transformations add to the algorithmic toolbox for pose-invariant shape analysis. Mathematically speaking, mean curvature is a shape invariant and in general fully characterizes closed shapes (together with the metric). Computationally speaking, spin transformations make that relationship explicit. Our work expands on a discrete formulation of spin transformations. Like their smooth counterpart, discrete spin transformations are naturally close to conformal (angle-preserving). This quasi-conformality can nevertheless be relaxed to satisfy the desired trade-off between area distortion and angle preservation. We derive such constraints and propose a formulation in which they can be efficiently incorporated. The approach is showcased on subcortical structures.