Representing rational curve segments and surface patches using semi-algebraic sets
In the work for generating surfaces by (piecewise) rational curves or surfaces, the representation of the surfaces (or curves) using semi-algebraic sets is an important tool. Such surfaces have many useful applications and there are also other methods to define them, such as radial basis functions. In this particular approach, rational curves or surfaces defined by tensor-products of rational curves are used. The main idea is the generation of globally defined functions whose local parts (segments) the desired surfaces are (in other words, the parts of the surfaces we really need are sections of the unbounded ones). They are defined implicitly by semi-algebraic sets, and the provided algorithms for computing the surfaces work well unless there are singularities in the rational function. These singularities need to be identified, and this is achieved by creating boxes about the singularities where the curves are generated within a certain numerical error (by contrast, there is no error if there is no singularity). Many numerical examples are provided to show the success of the methods.
scientific article; zbMATH DE number 7137425
scientific article