Figure 4.5 Application of mirror inversion to cylindrical and tangent surfaces.
the mirrored half. These are the only impact on the crease pattern. If the original shape can be made with one sheet, then so can the mirrored shape. Examples in Figure 4.5 use a cylindrical surface and a tangent surface (touching a helix) as the first shape, respectively. Mirror inversion on a developable surface several times produces a variety of shapes containing curved folding.* Note that all the curves made here are planar curves (curves resting on a plane) and not spatial curves.
4.3 Specifying Mirror Planes by a Polygonal Line We have seen so far that 3D folding is achieved by repeatedly applying mirror inversion to a one-sheet shape. This section shows how mirror inversion is applied by specifying a polygonal line. For handiness, the original shape is limited to a “cylindrical surface.” The cylindrical * At this writing, the ORI-REF software for assisting mirror inversion is available to the public at the following URL: http://mitani.cs.tsukuba.ac.jp/ori_ref/ (accessed on March 7, 2016).
surface here is obtained by sweeping one polygonal or curved line in one direction and consists of a set of parallel straight lines. In Figures 4.6 and 4.7, the bold solid polygonal line is called the section line and the arrowed sweep direction is called the sweep locus. The sweep locus is perpendicular to the plane where the section line rests. The cylindrical surface thus obtained consists of a set of rectangles and can be made by folding or bending a sheet. Now, apply mirror inversion on this shape to make a new shape. Hereafter, the shape before the folding operation as in Figure 4.6 is called an initial developable surface. Deciding the mirror plane position for this initial developable surface determines the shape obtained by mirror inversion. Several folding operations are achieved by placing multiple mirror planes. When a part of a solid is flipped by a mirror plane, the sweep locus is flipped together. Then, the sweep locus travels as it bounces off the mirror plane, as in Figure 4.7a. If the plane, where the section line rests, and mirror surfaces are placed vertically, the sweep locus keeps traveling horizontally. It goes like a light beam reflecting from a mirror at the same incident and reflection angles, as Figure 4.7b shows. Making Use of Mirror Inversion
K29074_Book.indb 53
53 5/12/16 12:54 PM