Subdivision Of Non-uniform B-Spline Curves Based On Blossoming

As the method of subdivision possesses the advantages of simplicity, high efficiency andapplicability to arbitrary topology, it is welcomed by graphics scholars and has been the inter-national research focus in the field of Computer Aided Geometric Design(CAGD) and Com-puter Graphics(CG). Subdivision method is a way of refining grids repeatedly based on certainrules in order to achieve a sequence of grids are claimed to converge to a limit, which is thesmooth curve or surface. With the blossoming method, this thesis studied binary and ternarynon-uniform, refine and smooth subdivision algorithm in order to meet the symmetry and re-ducibility. The following three aspects are the main results of this thesis:Motivated by the construction for non-uniform subdivision algorithm with reducibility, wedefine the Double-d non-uniform subdivision algorithm and present the necessary and sufficientconditions on subdivision algorithm with symmetry or reducibility. Double-2Both subdivisionalgorithms and Double-3Both subdivision algorithms with symmetry and reducibility are con-structed based on blossoming. And Double-5Symmetric subdivision algorithm is constructedtoo. In the meantime, we compare with other symmetric and reducible algorithm. And thevalidity of the algorithms are verified. This thesis present the necessary and sufficient conditionon non-uniform, refine and smooth subdivision algorithm with symmetry through relaxing theconditions of Double-d subdivision algorithm.The above is a study on the binary. This thesis present Triple-2Symmetric subdivisionalgorithm and Triple-3Symmetric subdivision algorithm based on blossoming in the Triple-dnon-uniform subdivision algorithm. By comparison, we know that the growth rate of the controlvertices of ternary non-uniform subdivision algorithm is fast, and ternary non-uniform subdivi-sion algorithm has higher approximation and smoother result. Ternary non-uniform subdivisionalgorithm are more suitable for practical problems and can get better result of subdivision afterfewer steps of refinement for complex curves. The above research of refine and smooth sub-division algorithm of B-spline for d degree rich and perfect the research on the symmetry andreducibility of the subdivision scheme.This thesis present a non-uniform subdivision algorithm for B-spline of arbitrary degreebased on blossoming through relaxing the conditions of Double-d subdivision algorithm. Wecompare this new subdivision algorithm with other algorithms and analyze the difference on theresults and calculations. We get the conclusion that the process is different, but the result is thesame. The calculations of this algorithm is close to the existing algorithms when ignoring the influence of degree. We give a detailed proof of the correctness of this algorithm by constructingtwo polynomials of blossoming. At the same time, we improve this algorithm to be understoodand implemented easily based on the purpose of reducing the amount of storage and unifyingodd and even.