Spherical Geometrical Bases of Spherical Origami

TL;DR

This paper develops a comprehensive geometrical framework for spherical origami, extending Euclidean origami concepts to spherical geometry and introducing new fold curves for 3D spherical folding.

Contribution

It formalizes spherical origami using spherical geometry, extending axioms and introducing equidistant curves for 3D folding, with practical computer graphics applications.

Findings

All seven Huzita--Justin axioms have explicit spherical equations.

The framework enables construction of spherical origami models like birds.

Validated through computer graphics demonstrating theoretical and practical utility.

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.