On the Interconnection of a Range and the Second-Order Cluster

Abstract

In accordance with “Specialized sections of affine, projective and computational geometry” syllabus for Master’s degree program in “Multimedia systems and computer graphics” developed at the Far Eastern State Transport University, the subject “Projective theory of the second-order curves” is considered [4; 14; 18]. Both at the sources mentioned and the textbook [11] projective method of the second-order curves formation as a range of the second order and its dual form – a second-order cluster (with regard to well-known theorems and relations, including Pascal and Brianchon theorems) is discernible. However, the graphical interpretations represented at the sources mentioned have general abstract character: to form the secondorder range two projective clusters of the first-order with the corresponding right lines are defined, and to design the second-order range – two projective series with the corresponding points. Techniques of high value can be observed when constructing outlines with the second-order curves; in this case, depending on engineering discriminant values, these curves can be constructed both using Pascal lines and qualities of the engineering discriminant itself, that is paying attention to the fact that tangents to the second-order curves makes the second-order cluster. Naturally, intent arises not to set the corresponding points on projective ranges, but to get them by elaboration, disclosing upon that regularities when constructing different second-order curves (the first aspect of research). The second aspect is in the consider - ation of the particular cases which would have definite secondorder clusters. In this case the task would be to model the secondorder range as a dual form of cluster. Thus it would be possible to get the interconnection of the definite cluster and the second-order cluster.