Classification Of Quadratic Parametric Curves And Surfaces Based On ?-bases

The quadratic curves and surfaces are common models in geometric modeling.The usually expression of curves and surfaces are parametric form and implicit form.In recent years,a new type of algebraic tool developed from the theory of moving curve and surface,?-basis,can complete the mutual conversion between parametric and implicit equation.So ?-basis can be seen as a bridge between its parametric form and implicit form,for this kind of characteristic,we try to apply the ?-basis to the classification of quadratic curve and surface.That is,only using the parametric equations,the properties of ?-basis are directly used to classify the quadratic curve or surfaces,It avoids a tedious process of the parameter form to implicit form of curves and surfaces.For the classification of quadratic parametric curves,the basic knowledge of ?-basis is first summarized.Then,using the explicit structure of ?-basis to construct a discriminant,and the quadratic parametric curves is classified by the discriminant signs.In contrast,the classification of quadratic parametric surfaces is more complex.Fistly,the implicit times needs to be identified.For the quadratic parametric surfaces,we use the undetermined coefficient method to obtain the expression of ?-basis.Then using linear transformation to simplify the equations of quadratic surfaces.And combining the idea of traditional eigenvalue classification,we obtained the method based on ?-basis for classifying quadratic parametric surfaces.From the analysis results,the process of judging the type of quadratic curves and surfaces is simple and convenient for operation.Which overcomes the difficulty of calculation rely on the implicit equation in the traditional method.Moreover,this paper has achieved both the theoretical and practical effects.