Developing surfaces

Abstract

To transform a three-dimensional object or to map texture to its surface, it is necessary to introduce a coordinate system. If the surface can be cut and developed, it is easy to identify each point on the surface with the coordinate values. According to a theory in topology, any closed polygonized two-dimensional surface can be represented by a canonical development. However, no efficient algorithm to actually develop a given surface has been presented, and the theory sounds abstract. This paper proposes a method to develop an arbitrary polygonized closed surface and to establish the correspondence between each point on the surface and a point on a regular polygon. Educational software is developed using the algorithm that visualizes the coordinate system by texture mapping or by allowing a user to paint on the surface.

Flexible Developable Surfaces

“…The second goal is to approximate given geometric data, such as a scattered point clouds [CLL*99, Pet04, PB07, TC09, MS11], sets of approximating planes [PW99], or a nearby non‐developable surfaces; the latter may require segmenting the input surface into patches [Wan04, JKS05, STL06, Wan08, SKKO02] or strips [MS04, LLH09, PSB*08] and treating the case of curved creases [KFC*08].…”

“…Exact discretizations can also be developed by augmenting more general discretizations with developability constraints , an approach considered for Bézier and B‐spline surfaces [CC90, FB93, Pot95, CS02, Aum03, Aum04, FJ07], triangle meshes [WF88, Fre02, Fre04, Wan04, MS04, BGW06, RSW*07, LTJ07, LLH09, CT10, RCHT11, SKKO02], and planar quadrilateral meshes [WF88, LPW*06, KFC*08]. The majority of these discretizations have been demonstrated successfully in the context of synthesis and approximation, but their performance in problems of smooth deformation remains less understood.…”