Spherical images and inextensible curved folding.

Area of Science:

• Mechanical Engineering
• Applied Mathematics
• Geometry

Background:

• Duncan and Duncan (1982) analyzed inextensible surfaces folded about a curve, deriving relationships between generators, path shape, and fold angles.
• Two special cases were presented for simplified folding in sheet-metal processes: uniform fold angle and planar folded curves.

Purpose of the Study:

• To verify the special cases presented by Duncan and Duncan using a graphical method.
• To provide a new geometric approach for analyzing developable folding patterns.
• To enable the synthesis of novel deployable and shape-changing surfaces.

Main Methods:

• Graphical treatment using a method of Gauss.
• Replacing the fold curve with a piecewise linear path connecting hinge line vertices.
• Constructing spherical images for developable folding of successive vertices, inspired by d-cone analysis.

Main Results:

• The geometric relationships between spherical images directly reveal the operating conditions of Duncan and Duncan's special cases.
• The proposed method offers a straightforward verification and understanding of folding patterns.
• The approach simplifies the analysis of folding for practical applications.

Conclusions:

• The graphical and spherical image method effectively verifies and clarifies the special cases of inextensible surface folding.
• This approach facilitates the design and synthesis of complex deployable and shape-changing surfaces.
• The method avoids complex calculations, making it accessible for broader engineering applications.