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. Given an initial design, the proposed method finds a triangulated surface with user-specified Gaussian curvatures (not limited to constant Gaussian curvatures) while keeping some of the vertices fixed. 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. This allows the free-form design of discrete architectural surfaces that achive curvature requirements posed by stiffness and constructability. Furthermore, controllability on the conformal class suppresses possible distortion of the panels, resulting in higher structural performance and aesthetics. Our method relies on the idea in discrete differential 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. Due to this difficulty, few concrete applications of the discrete Ricci flow have been reported which specifically aims at creation of architectural surfaces. We propose a perturbation of the discrete Ricci energy and develop a least-squares-based optimisation scheme to address these problems with a working implementation available online.