# Infinitesimal conformal deformations of triangulated surfaces in space

## TL;DR

This paper investigates infinitesimal conformal deformations of triangulated surfaces in Euclidean space, linking geometric changes to dihedral angles and stereographic projections, advancing understanding of surface flexibility.

## Contribution

It introduces a new parametrization of infinitesimal conformal deformations via dihedral angles and explores their relation to stereographic images and the Schl"{a}fli formula.

## Key findings

- Deformations preserve length cross-ratios.
- A one-to-one correspondence exists with stereographic projections.
- Deformation space relates to dihedral angle changes.

## Abstract

We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand, conformal deformations generalize deformations preserving edge lengths. On the other hand, there is a one-to-one correspondence between infinitesimal conformal deformations in space and infinitesimal isometric deformations of the stereographic image on the sphere. The space of infinitesimal conformal deformations can be parametrized in terms of the change in dihedral angles, which is closely related to the Schl\"{a}fli formula.