Planar Polygon Deformation Algorithm

Morphing, also known as metamorphosis, is the gradual, consecutive, slippery and natural transformation of the source object into the target object (the object maybe digital image, curve surface, gridding and so on). Morphing has wide practical use in areas such as computer graphics, animation design, industrial modeling, science computation visualization, film stunt and so on. This paper makes researches on the morphing of planar compatible triangulations and planar polygons, and the main results are as follows:1) The similitude compatible triangulations of planar polygons. This paper presents a compatible triangulations arithmetic which is based upon the comparability between the source polygon and the target polygon. This arithmetic takes the comparability into account. Firstly, triangulates the similitude parts between the source polygon and target polygon, needing no extra vertex in this process, this can simplify the two polygons, called the two resulted polygons as simplified source polygon and simplified target polygon. Using the already existing arithmetic, the two simplified polygons can be triangulated. Then the computation can be decreased for triangulating the original polygons and the results of triangulation are satisfied. The number of extra vertexes can be decreased, so the complexity and the computation are decreased for metamorphosizing the polygons.2) The morph arithmetic of polygons is based upon waveletes. Firstly, the source polygon and the target polygon are decomposed at the same level, as a result, the overall shapes of the polygons and the details of the polygons can be gotten. Secondly, the overall shapes and the details are metamorphosized separately. Lastly, the middle polygon can be reconstructed via reconstructed arithmetic according to the middle overall shapes and the middle details. The intersection-free of the overall shape of the polygons is largely improved, so this arithmetic can be merged with the morph arithmetic of the compatible triangulation, to exert the virtues of the two arithmetices. There are some examples to show that this arithmetic can decreased the complication of compatible triangulation, bringing on the decreased the computation of morph andthe process is real time. The morph resultes are satisfied.3) This paper goes along with the research on convexity-preserving when planar convex griddings metamorphosize. When two polygons that have different convex boundaries metamorphosize, the arithmetic presented in this paper can keep the boundary convexity. This arithmetic is base on the angle and that is used to metamorphosize the boundaries of the griddings. The character that this arithmetic is convexity-preserving is proved in this paper and this arithmetic has the character that all the angles in the polygon have the same transformational rate. The inner vertexes can be metamorphosized via convex combination. The arithmetic in this paper can keep the boundary protruding all the time and the middle gridding is compatible with source gridding and target gridding, the phenomena of intersection will not appear in the whole process of metamorphosise.