An approach to automating the construction of 3D girihs (3D Islamic geometric patterns) on NURBS surfaces

Islamic geometric patterns known as “girihs” are typically seen as 2D designs. However, in traditional Iranian architecture, girihs have been used to decorate curved and 3D surfaces. While there have been many studies on 2D girihs and methods for creating them, there has been less focus on 3D girihs. Additionally, there isn’t a comprehensive digital technique for constructing 3D girihs. This paper aims to fill this gap by proposing an automated approach for constructing girihs on curved surfaces to create 3D girihs. We use the UV coordinate relative to the NURBS (Non-Uniform Rational B-Spline) surfaces to identify the corresponding generative points of 2D girihs on 3D surfaces, then connect the points to construct 3D girihs. With this method, any geometric girih can be mapped onto a 3D surface, and under the right conditions, any 2D designs and motifs can also be mapped. We implement the proposed method in Grasshopper for Rhino 8 for Win, controlling the density and mapping quality of 3D girihs through parameters N MU and DL. By integrating this algorithm with a 2D girih generation algorithm, we provide the possibility of real-time customization and interactive construction of 3D girihs.