Free-form Design of Discrete Architectural Surfaces by use of Circle Packing

This paper presents an efficient approach for the conceptual design of architectural surfaces which are composed of triangular panels. In the free-form design of discrete architectural surfaces, the Gaussian curvature plays an important role not only aesthetically but also in terms of stiffness and constructability. However, designing a surface manually with specific Gaussian curvatures can be a time-consuming task. We propose a method to find a triangulated surface with user-specified Gaussian curvatures (not limited to constant Gaussian curvatures) and boundary vertex positions. In addition, the conformal class of the final design can be specified; that is, the user has control over the shape (the corner angles) of each triangular panel. The panels could be encouraged to form a regular tessellation or kept close to those of the initial design. The controllability of the conformal class suppresses possible distortion of the panels, resulting in higher structural performance and aesthetics. Our method relies on the idea in computational conformal geometry called circle packing. In this line of research, the discrete Ricci flow has been widely used for surface modelling. However, it is not trivial to incorporate constraints such as boundary locations and convexity of the spanned surface, which are essential to architectural applications. We propose a perturbation of the discrete Ricci energy and develop a least-squares-based optimisation scheme to address these problems with an open-source implementation available online.