🤖 AI Summary
This study addresses the absence of a rigorous geometric framework for spherical origami, particularly concerning folding rules on the unit sphere and their realization in three-dimensional space. The work proposes the first complete axiomatic system for spherical origami by extending the classical Huzita–Justin axioms from Euclidean to spherical geometry, providing explicit equations for all seven axioms. Innovatively, it replaces geodesics with isometric curves as three-dimensional creases, thereby expanding the repertoire of realizable folded forms. By integrating spherical and differential geometry with computational origami theory and computer graphics, the authors successfully generate a spherical origami bird model, demonstrating both the theoretical completeness and practical feasibility of the proposed framework.
📝 Abstract
This paper establishes a rigorous geometrical framework for spherical origami, origami using spherical sheets based on spherical geometry. Two settings are treated: origami restricted to the unit sphere ($\mathbb{S}^2$), and three-dimensional folding of spherical sheets in space. For origami on $\mathbb{S}^2$, the definitions of Euclidean origami are systematically extended to the spherical setting, and all seven Huzita--Justin axioms are shown to admit explicit equations in spherical geometry. For three-dimensional folding, equidistant curves are introduced as fold curves, replacing geodesics and enabling a richer family of folds. The framework is validated by successfully constructing computer graphics of spherical origami birds, demonstrating both the theoretical completeness and practical utility of the proposed approach.