Algebraic Structure Of Some Conic Curves And Pascal Lines' Track

Pascal lines are pretty important in planar algebraic curves. And planar algebraic curve is an important research object and tool. Furthermore, people found that the relationship between intrinsic properties of planar algebraic curves and the singularity ofbivariate spline spaces is m-ore and more important.Luo and Chen[7] established an equivalence relation between the study of the intrinsic properties of planar algebraic curves and the study of singularity of splme space. And they found an intrinsic inyariant of any algebraic curve by means of spline approach. So, We can apply some methods and conclusions of one hand to the other hand. Therefore, the property, of Pascal line is important for both algebraic curve and singularity of spline spaceThe aim of this paper is to find common properties among conic curves which have the same Pascal line and compute the track of Pascal lines. We convert a conic curve to the relations-hipof points and lines. By introducing a new product between point and point and between line and line, we define a new product among conic curves. By the new product, we make these conic curves to be a hemigroup. And the construction of the hemigroup is also investigated, which is that the hemigroup can be divided into several sub-semigroups which are identical to each other. We also consider two geometry modelsand compute Pascal lines' tracks according to these geometry models. The two geometry models can present the general cases.