This paper proposes a design method to obtain a family of rigidly foldable structures with one degree-of-freedom (DOF). The mechanism of flat-foldable degree-four cones and mutually compatible cones sharing a boundary is interpreted as the mechanism of Bricard's flexible octahedra. By sequentially concatenating compatible cones, one can design horn-shaped rigid-origami mechanisms. This paper presents a method to inversely obtain rigidly foldable horns that follow given space curves. The resulting rigidly foldable horns can be used as building blocks for a transformable cellular structure and attachments to existing rigidly foldable structures.