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research-article

Gauss Map Based Curved Origami Discretization

[+] Author and Article Information
Liping Zhang

Dept. of Mechanical Engineering, Dalian Polytechnic University, Dalian, China 116034
lipingzhang3@163.com

Guibing Pang

Dept. of Mechanical Engineering, Dalian Polytechnic University, Dalian, China 116034
pangguibingsx@163.com

Lu Bai

Dept. of Mechanical Engineering, Dalian Polytechnic University, Dalian, China 116034
bailu0725@163.com

Tian Ji

Dept. of Mechanical Engineering, Dalian Polytechnic University, Dalian, China 116034
ji_t@163.com

1Corresponding author.

ASME 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 satisfying the equivalent surface curvature changement discretizations. Solving basic folding units occurs in such tasks as simulating the behavior of Gauss mapping. This paper starts from the curved folding unit analysis which at least has a curved folding line. A curved folding line connects two different curved developable surfaces by transferring its curvature change. Gauss map is utilized to investigate the normal curvature change of the curved surface. The Gauss spherical curves of different developable surfaces are set up under the Gauss map. In this way, spatial curved surfaces are mapped to spherical curves. Each point on the spherical curve represents a normal direction of one 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. While this normal vector direction can be mapped back to identify the specific ruling lines which can be a discretization of the curved surface. These developable curved surfaces are then discretized into planar patches to acquire the geometric properties of curved folding such as folding angle, folding direction, folding shape, rigid foldability and geometric constraints of adjacent ruling lines etc. It acts as a connection of curved and straight folding knowledge. The approach described in this paper is applicable to any generic developable surfaces. The utility of the technique is demonstrated with the principles of Gauss spherical curves.

Copyright (c) 2018 by ASME
Topics: Shapes
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