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Research Papers

Gauss Map Based Curved Origami Discretization

[+] Author and Article Information
Liping Zhang

Department of Mechanical Engineering,
Dalian Polytechnic University,
Dalian 116034, China
e-mail: Lipingzhang3@163.com

Guibing Pang

Department of Mechanical Engineering,
Dalian Polytechnic University,
Dalian 116034, China
e-mail: pangguibingsx@163.com

Lu Bai

Department of Mechanical Engineering,
Dalian Polytechnic University,
Dalian 116034, China
e-mail: bailu0725@163.com

Tian Ji

Department of Mechanical Engineering,
Dalian Polytechnic University,
Dalian 116034, China
e-mail: ji_t@163.com

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received September 24, 2017; final manuscript received September 26, 2018; published online November 13, 2018. Assoc. Editor: Jian S. Dai.

J. Mechanisms Robotics 11(1), 011006 (Nov 13, 2018) (11 pages) Paper No: JMR-17-1325; doi: 10.1115/1.4041631 History: Received September 24, 2017; Revised September 26, 2018

Abstract

This paper addresses the problem of discretizing the curved developable surfaces that are satisfying the equivalent surface curvature change discretizations. Solving basic folding units occurs in such tasks as simulating the behavior of Gauss mapping. The Gauss spherical curves of different developable surfaces are setup under the Gauss map. Gauss map is utilized to investigate the normal curvature change of the curved surface. In this way, spatial curved surfaces are mapped to spherical curves. Each point on the spherical curve represents a normal direction of a ruling line on the curved surface. This leads to the curvature discretization of curved surface being transferred to the normal direction discretization of spherical curves. These developable curved surfaces are then discretized into planar patches to acquire the geometric properties of curved folding such as fold angle, folding direction, folding shape, foldability, and geometric constraints of adjacent ruling lines. It acts as a connection of curved and straight folding knowledge. The approach is illustrated in the context of the Gauss map strategy and the utility of the technique is demonstrated with the proposed principles of Gauss spherical curves. It is applicable to any generic developable surfaces.

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Figures

Fig. 1

Different origami states: 3D folded surface, 2D curved folding crease, polyhedral pattern, and discretized crease

Fig. 2

Developable surface

Fig. 3

Gauss map

Fig. 4

Developable surface and the Gauss spherical curve

Fig. 5

A curved-folding unit and its Gauss spherical curves

Fig. 6

Multiple curved lines have same Gauss spherical curve

Fig. 7

Gauss spherical curve and the fold angle (a) geometric analysis of point P and (b) geometric relations of spherical curves

Fig. 8

Infinite continuous big circles between ⊙P and ⊙Q

Fig. 9

Tangent vectors and the normal planes

Fig. 10

Tangent surface discretization: (a) a tangent surface and its ruling lines, (b) setting up continuous normals, (c) Gauss map, (d) discretization of spherical curve, (e) the discretized normals, and (f) the resulted discretized ruling lines

Fig. 11

Curved folding unit discretization: (a) a curved folding unit, (b) Gauss map curves, (c) the discretization of spherical curves, (d) the discretized points on spherical curves, (e) the discretized normals, and (f) the discretized ruling lines

Fig. 12

Geometric relations of the curved fold and curved surface discretization

Fig. 13

Fold angle of the curved folding surfaces

Fig. 14

Fold angle between the discretized polyhedral patches: (a) a curved folding unit, (b) the discretized folding unit, and (c) fold angle of the discretized patches

Fig. 15

Discretization approximation of tangent surface: (a) a curved tangent surface, (b) its approximation polyhedron, (c) spatial skew quadrilaterals, (d) fold angle βM between two patches, and (e) unfolded skew quadrilaterals

Fig. 16

Discretization error analysis: (a) normals of curved surface and its approximation polyhedron and (b) error metric

Fig. 17

Error analysis of (a) cylinder surface, (b) conic surface, and (c) tangent surface

Fig. 18

The parameters of an oblique cone surface

Fig. 19

The Gauss spherical curve of oblique cone surface

Fig. 20

Discretization of Gauss spherical curve

Fig. 21

Mapping back to oblique cone surface discretization

Fig. 22

Error analysis graph of oblique cone surface discretization

Fig. 23

Directrix generation

Fig. 24

Tangent developable surface construction

Fig. 25

Gauss spherical curve generation

Fig. 26

Discretization of Gauss spherical curve

Fig. 27

Mapping back to discretize tangent surface

Fig. 28

Error analysis graph of tangent surface discretization

Fig. 29

The potential mechanism model of a discretized origami: (a) a discretized origami, (b) its mechanism model, (c) constructive spherical mechanism, and (d) network of spherical mechanisms

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