Free-form Surface Approximation Using Rotational Patches

We present a method to approximate free-form surfaces using assemblies of rotational patches for architectural rationalization. Rotational surface patches inherently allow for the simultaneous repetition of multiple building elements along the arc direction. By assembling multiple patches, we can create diverse free-form-like geometries to satisfy broad design intents, while preserving local symmetry to enable cost-effective element fabrication. The main challenge lies in the strict constraint of maintaining local rotational symmetry, while ensuring the final tessellated form is seamless, smooth, and closely resembles the target surface. To address this, we propose a patch layout creation approach that segments the input surface into patches, resembling untrimmed rotational patches within a prescribed error threshold. Additionally, we develop a B-spline-based optimization framework to refine the fitted rotational patches for smooth connections and faithful surface approximation. To facilitate practical architectural applications, we provide a post-processing tool that converts the discrete patch assembly into a seamless, smooth quad mesh composed of locally repeated elements. We demonstrate that our approach is applicable to a variety of free-form surfaces, including those that mimic iconic architectural designs, and can address various practical requirements for a wide range of application scenarios.