Shape Modelling and Approximation with Geometric and Fabrication Constraints

Abstract

Fabrication-aware design plays an important role in the modern manufacturing-related industry. Fabrication-aware design introduces innovative tools and methodologies enhancing various aspects of manufacturing processes, including design, prototyping and mass production. These novel methods are widely applied in industries such as artistic craft design, consumer product manufacturing and architectural construction. Designing and producing with fabrication-aware constraints can reduce time and labour costs while minimising material waste. With the development of 3D fabrication technologies, such as CAD and CAM, the application of fabrication-aware design on modelling and approximation of 3D shapes has attracted attentions of both academics and industries. It is also worth exploring the potential of combining fabrication-aware design with 3D geometry algorithms. This thesis contributes to the topic of fabrication-aware shape modelling and approximation. It addresses several challenges in this area and presents solutions which take both geometric and fabrication constraints into account. In particular, it will present approaches for designing pop-up paper structures, modelling special piecewise developable surfaces, and approximating Weingarten surfaces. Paper crafts, such as popup structures and origami, contain various 2D and 3D geometric properties. Thus the design and manufacturing of these paper crafts is an active research topic in the fields of computer graphics and mathematics. Traditionally, manually designing paper crafts with pop-up structures, is a time-consuming process involving trial and error. A novel optimisation-based approach will be introduced to improve the efficiency of designing and modelling pop-up paper structures. The developable surface is a widely used geometric structure in product and architectural design. The D-Form is a special piecewise developable surface containing two pieces of developable surfaces and one seam. An approach will be presented to generate D-Form from either two given planar domains or from a given space curve. Principal curvatures of a Weingarten surface have a functional relation. This property benefits the design of architectural surfaces through mold re-use. However, approximating a given surface by Weingarten surfaces is a challenging problem. A mesh-based approach will be proposed to approximate Weingarten surfaces by iteratively performing principal curvature diagram transformations and shape deformations. We will discuss the background, methods, limitations and future work of each approach in this thesis. We also demonstrate these approaches with various examples.

Date of Award	25 Jun 2025
Original language	English
Awarding Institution	University of Bath
Supervisor	Yongliang Yang (Supervisor) & Xi Chen (Supervisor)