C To A Continuous Adjustable Subdivision Surfaces

Subdivision surfaces arc gotten by iterative subdivision algorithms from step discretion to step discretion, which avoid continuous step. Because the idea of subdivision algorithm is simple and convenient to implement, subdivision algorithm has been a focus in computer aided geometric design and computer graphics recently. There has been a perfect subdivision system, such as Catmull-Clark subdivision, Loopsubdivision, Butterfly subdivision, 3^1/2 subdivision, 2^1/2 subdivision and hybridsubdivision etc. Once the initial mesh is given and the subdivision algorithm is chosen, the final subdivision modeling that isn't modified is decided. Considering theshortcoming of subdivision algorithms, we study C¹ continuous modified subdivision algorithm based on triangular meshes, quad meshes and hybrid meshes.Subdivision algorithm is divided into three steps: linear interpolation, average and correction. In this process, parameters λ and ω are introduced in this step to improve the freedom of modeling, so the shape of modeling is modified by changing parameters λ and ω. The involved three problems are:(1) modified subdivisionalgorithm of triangular meshes, that is ,we can got more subdivision surfaces byintroducing modified parameters λ ,and when λ=3/8, typical Loop subdivisionsurface is gotten. (2) modified subdivision algorithm of quad meshes, that is, theshape to surfaces is modified by adding parameter ω , and when ω = 1/4,we can getfamous Catmull-Clark subdivision surface. (3) modified subdivision algorithm of hybrid meshes, that is , two parameters A and co produced in the average process can be chosen to modify surfaces, so we can get colorful surface models. When A and co are given special values, we can separately get Loop subdivision on triangular meshes and Catmull-Clark subdivision on quad meshes. Here, parameters A and co have specific geometric meaning. The algorithm is simple and convenient to implement. By it we can get both smooth models and special ones, which rich thetype of subdivision modeling.In this paper, we separately give sufficient conditions for C continuous modified subdivision algorithm and prove them based on different meshes. Finally, some cases are given to illustrate the efficiency of the algorithm. Jn addition, we use simple data structures such as matrix and list instead of complicated data structures.