Triangle center

Abstract: The paper aims at introducing novel distance measures for the intuitionistic fuzzy set (IFS) to discuss the decision-making problems. The current work exploits four different notions of centers, namely centroid, orthocenter, circumcenter and incenter of the triangle. First, we mold knowledge embedded in IFSs into isosceles TFN (triangular fuzzy number). Hence, based on these TFNs, we design four-novel distance/similarity measures for IFSs using the structures of the four aforementioned centers and inspect their properties. To avoid the loss of information during the conversion of IFSs into isosceles TFNs, we included the degree of hesitation (t) between the pairs of the membership function in the process. The compensations and authentication of the proposed measures are established with diverse counter-intuitive patterns and decision-making obstacles. Further, a clustering algorithm is also given to match the objects based on confidence levels. The performed analysis shows that the proposed measures give distinguishable and compatible results as contrasted to existing ones.

Abstract: In three-dimensional graphics rendering, a method of texture mapping, or shading, applies to triangle-based graphical objects having undergone a perspective transformation. The present invention makes use of linear interpolation for determining the appropriate mapping for the interior points of each triangle, thus reducing the computation-intensive mathematical calculations otherwise required. In order to minimize visual artifacts due to high interpolation errors, the borders of each triangle are tested against a predetermined threshold, and the triangle subdivided if any of the borders contain a maximum error which exceeds the threshold. The subdivision continues until all triangle sides have maximum errors that are less than the threshold value. Linear interpolation is then used to determine all mappings for the sides and interior points of the triangle. In alternative embodiments, the triangle is subdivided without using recursive methods. In one non-recursive method, the entire triangle is subdivided uniformly based on the necessary number of segments into which the triangle sides must be broken to keep the maximum error below the threshold. In another non-recursive method, w-isosceles triangles are subdivided into trapezoids, each of which is then subdivided into w-isosceles, and mostly geometrically isosceles, triangles.

Abstract: We investigate the geometric properties of simplices in Euclidean d-dimensional space for which two or more of the analogues of the classical triangle centers (including the centroid, circumcenter, incenter, orthocenter or Monge point, and the Fermat-Torricelli point) coincide. We also investigate the geometric significance of the cevian line segments through a given center having the same length. We give a unified presentation, including known results for d=2 and d=3.

Abstract: We consider a kind of problem that appears to be new to Euclidean geometry, since it depends on an understanding of a point as a function rather than a position in a two-dimensional plane. Certain special points we call centers, including the centroid, incenter, circumcenter, and orthocenter. For example, the centroid, as a function of the class of triangles with sidelengths in the ratio a1 : a2 : a3, is given by the formula 1/a1 : 1/a2 : 1/a3. The kind of problem introduced here leads to functional equations whose solutions are centers.