Developing Surfaces

To transform a three-dimensional object or 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.