Planar Polygon Deformation Technology

Morphing, also knows as metamorphosis, is the continuous, smooth and natural transformation of a source object into a target object, where the object can he digital images, polygons, freeform curves and surfaces, meshes, polyhedrons etc. Along with the speedy development of computer technology, morphing is becoming popular in many areas, such as computer graphics, industrial design, computer animation, virtual reality and film stunt. Shape morphing of planar polygon, that is shape blending, is also attractive, which has great significance in key frame animation, pattern recognition, surface reconstruction and three-dimensional modeling.In this thesis, we introduce the history and the current development in the area of planar polygon morphing concisely, also point out respectively the characteristics and its insufficiencies of the main methods in this research area as well as its open questions. In the morphing process, how to avoid edge self-intersection and preserve the geometrical properties of the intermediate shapes are two challenges. Several fruits on these problems are gained by the former researchers, but these seems insufficient, we propose an efficient approach which has low computational complexity, and is edge self-intersection-free and can preserves the geometry better in morphing process. In this method presented in this paper, we first determine an affine transform as rigid transformation by least square principle so that the resulting polygon after the transformation being applied on the source polygon is as close as possible to the target polygon in the whole, which preserves the geometry of the intermediate polygon morphing more uniformly. Then we embed the source polygon and target polygon into compatible planar triangulation and express their vertices by mean value coordinators respectively, so the shape blending in detail pan is performedby morphing the corresponding compatible triangulations.In this paper, based on the edge vector representations of planar polygon, we also present a new method for polygon morphing by interpolating the corresponding edge vectors and their rotation matrixes of the initial polygon and the target polygon. This algorithm is simple and has low computing complexity, and runs fast that can be done in time.Several examples show our algorithms are valid and feasible, and natural morphing effects are performed.