  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.
