Isomorphic Plane Triangular Grid And The Planar Polygon Deformation

Morphing is the continuous smooth and natural transformation of a source object into a target object, where the object can be a numerical image, curve, surface, mesh, etc. Morphing has very wide use in many areas, such as computer graphics, animation design, industrial modeling, science computation visualization, film stunt, etc. This paper makes researches on the morph of compatible planar triangulations and that of planar polygons, and the main results are as follows:1) Morph of compatible planar triangulations: This paper presents a convexity-preserving method for morphing compatible planar triangulations with different convex boundaries. This method combines the intrinsic solution algorithm and the convex combination morph algorithm, showing that the former has an excellent property of preserving the convexity of the intermediate triangulations' boundaries. It guarantees that the boundary polygons of the triangulations preserve convexity all the time during the morphing, and that the intermediate triangulation at any time is compatible with the source and target triangulations, i.e. free of self-intersection. At the same time it realizes a convexity-preserving morph of the two convex polygons.2) Intersection-free morph of planar polygons: This paper presents a shape feature based and triangulations embedded method for morphing planar polygons. This method embeds the source and target polygons in compatible planar triangulations whose boundaries are the magnified convex hull of polygons respectively, then morph the embedded triangulations with the convexity-preserving method for morphing compatible planar triangulations presented in this paper. In contrast with the method of Gotsman and Surazhsky, this method considers the geometric contours as well as the differences of the source and target polygons, so the morph is more natural. On the other hand, the algorithm in the method for compatible triangulation reduces the number of Steiner vertices greatly.3) Locally modifiable morph of planar polygons: This paper presents a methodfor morphing planar polygons via discrete curvature interpolation. This method employs the definition of discrete curvature of Carmel and Cohen, and presents the algorithm for morphing via discrete curvature interpolation. On the basis of this, a locally modifiable algorithm is given, which is greatly intuitive and enables users to modify the polygons locally according to their requirement and desire preferably. The method is characterized by simpleness, availability, monotone change of total length of polygon sides and local modification.