Towards the suppression of blend chains using Euler operators and open geometric modeling kernel

This paper presents a CAD model simplification procedure which consists of recognition and suppression of certain types of blend chains. The proposed method involves Euler operators KEV, KEF, and KFMV, which are developed on top of open-sourced geometric modeling kernel. The simplification process employs two stages: recognition and suppression. The suppression stage ensures the geometric and topological validity of the simplification result. The present approach is targeted for use in a batch mode, which poses strict requirements to the robustness of the suppression algorithm. The essential properties of the presented approach are its sustainability, the predictability of the result and the extensible architecture, which allows for adding new topological cases without affecting the algorithm’s core. At the recognition stage, the algorithm constructs an attributed adjacency graph, which is then enriched with such information as edges’ types, their properties, and the assumed kinds of blend faces. At the suppression stage, the algorithm iterates the adjacency graph and composes the blend candidate faces into the chains. For each face in a chain, a local topology analysis is performed to determine the corresponding sequence of Euler operators which are supposed to eliminate that face. The algorithm allows for extensions through adding descriptors of the new topology situations into the processing. After the Euler operators are done, the affected edges are reconstructed to obtain a watertight boundary representation of the model.

Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators

Various investigators have studied the degree of approximation of a function using different summability (Cesáro means of order α: Cα, Euler Eq, and Nörlund Np) means of its Fourier-Laguerre series at the point x=0 after replacing the continuity condition in Szegö theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by (T·Eq)-transform of its Fourier-Laguerre series at frontier point x=0, where T is a general lower triangular regular matrix. A particular case, when T is a Cesáro matrix of order 1, that is, C1, has also been discussed as a corollary of main result.