# Smooth polyhedral surfaces

## TL;DR

This paper develops a theory for assessing the smoothness of polyhedral surfaces based solely on their geometry, introducing invariant notions inspired by differential geometry, with implications for architecture.

## Contribution

It proposes a novel framework for evaluating polyhedral surface smoothness using geometric invariants, bridging differential geometry and architectural design.

## Key findings

- Mild conditions restrict face shapes of smooth polyhedral surfaces
- Invariant notions of normal vectors and tangent planes are introduced
- The theory has potential applications in architectural geometry

## Abstract

Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the aim of our work is to find and to discuss suitable assessments of smoothness of polyhedral surfaces that only take the geometry of the polyhedral surface itself into account. Motivated by analogies to classical differential geometry, we propose a theory of smoothness of polyhedral surfaces including suitable notions of normal vectors, tangent planes, asymptotic directions, and parabolic curves that are invariant under projective transformations. It is remarkable that seemingly mild conditions significantly limit the shapes of faces of a smooth polyhedral surface. Besides being of theoretical interest, we believe that smoothness of polyhedral surfaces is of interest in the architectural context, where vertices and edges of polyhedral surfaces are highly visible.