Design support of a developable surface considering its bending energy

Abstract

A method to design a developable surface and its developed shape is proposed. A developable surface, which is generated by sweeping a straight line along a three-dimensional curve, can be seen many products such as ships, buildings, clothes, and so on. Although it is important to design the surface in case that designers give some constraints, its design process is not automated because of the difficulty of its modeling. Especially, cloth products are required to design not only their three-dimensional shapes but also their developed shapes. Therefore, we aim to propose a method to design the surface satisfying some constraints and its developed shape. As an example, we assume that some constraints are related to the control points of a curve in a developable surface. In this paper, it is revealed that the direction of a generatrix can be determined geometrically when a pair of three-dimensional curves are given. Moreover, in order to obtain a steadier shape, not only this condition but also the potential energy of the surface should be considered. From the above, we formulate optimization problems, upon this design process, which have different objective functions: one is the condition of a developable surface and the other is its bending energy. We compared and examined bending energies and maximum principal curvatures obtained by two different optimization problems. In order to verify our proposed method, we performed an experiment and confirmed its validity.

Keywords

• design
• modeling
• developable surface
• optimization
• automation