Calculation And Application Of Approximate ?-bases For Rational Curves And Surfaces

The technique of ?-bases is originated from the theory of moving lines and moving planes.Because of its special algebraic and geometric properties,it is an useful tool to study the curve and surface representation and related properties.In the field of geometric design,the approximate representation problem of rational curve and surface is one of the hot issues among many researchers for several decades.In this paper,we study the rational curve and surface by sparse approximate ?-bases,and use it to achieve the approximate representation.For the rational curve,we propose the concept of sparsity and define sparse ?-bases based on the approximate ?-bases.We determine the position of the base in the expression structure of sparse ?-bases by comparison of coefficient components.We give the algorithm of sparse ?-bases and optimize it.By optimizing the algorithm,we can get a simple representation of high-order approximate curve.Finally we introduce the concept of sparsity into the approximate implicit study of the rational curve.We give the algorithm of sparse approximate implicit form,and we can find the sparse approximate implicit equation that works well by the algorithm.For the rational surface,we choose to study the application of sparse approximate ?-bases in the ruled surface because that the high-order surface has complex structure.We define the sparse approximate ?-bases and give its algorithm.We discover the approximate effect is the best when the proportion of coefficient is maximum by comparing the examples.Therefore the reduction of approximate representation by sparse approximate ?-bases is a feasible new idea.