# Extending editing capabilities of subdivision schemes by refinement of point-normal pairs

## TL;DR

This paper introduces a 3D binary average for point-normal pairs to extend subdivision schemes, enabling more flexible geometry editing by refining initial normals, with practical implementation and demonstrations.

## Contribution

It develops a 3D point-normal averaging method and integrates it into classical subdivision schemes, enhancing their ability to generate and edit complex geometries.

## Key findings

- Modified schemes outperform linear variants in geometry editing.
- Normals can be effectively computed from meshes for initial data.
- Enhanced editing capabilities demonstrated through examples and videos.

## Abstract

In this paper we extend the 2D circle average of [11] to a 3D binary average of point-normal pairs, and study its properties. We modify classical surface-generating linear subdivision schemes with this average obtaining surface-generating schemes refining point-normal pairs. The modified schemes give the possibility to generate more geometries by editing the initial normals. For the case of input data consisting of a mesh only, we present a method for computing "naive" initial normals from the initial mesh. The performance of several modified schemes is compared to their linear variants, when operating on the same initial mesh, and examples of the editing capabilities of the modified schemes are given. In addition we provide a link to our repository, where we store the initial and refined mesh files, and the implementation code. Several videos, demonstrating the editing capabilities of the initial normals are provided in our Youtube channel.