Finetuning Discrete Architectural Surfaces by use of Circle Packing

This paper presents an algorithmic approach for the conceptual design of architectural surfaces represented by triangulated meshes. Specifically, we propose a method to optimise a surface according to user-specified geometric properties including the distribution of the Gaussian curvature and preferable boundary location. Designing a surface manually with specific Gaussian curvatures can be a time-consuming task, and the proposed method automates this task. Also, in the proposed approach, the resulting mesh could be encouraged to form a regular tessellation or kept close to those of the initial one. Our method relies on the idea in computational conformal geometry called circle packing and the discrete Ricch energy, which have been used for surface modelling. We develop a least-squares-based optimisation scheme by introducing a variant of the Ricci energy to accommodate flexibility in specifying design constraints such as boundary locations and convexity of the spanned surface, which are essential to architectural applications. We provide an open-source implementation of our method in Python. ¹¹ Our codes are publicly available at https://github.com/shizuo-kaji/ricci_flow.