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김성복 조선대학교 공학기술연구원 2022 공학기술논문지 Vol.15 No.2
This paper presents a systematic analysis of the signed barycentric coordinate concerning a triangle and its application to the smooth color shading of a square containing a triangle. The primary purpose of this paper is to establish a unified theory of the barycentric coordinate, which is valid regardless of whether a given point is inside, outside, or on the boundary of the referenced triangle. First, the barycentric coordinate in a triangle is geometrically defined as the ratio of the signed areas of three sub-triangles. Second, the algebraic computation of three components of the barycentric coordinate is described, which facilitates the efficient calculation of the signed values of the three components. Third, based on the sign combination of the barycentric coordinate, 2-dimensional space is wholly divided into a total of 19 regions, and the decision table is provided to determine the region where a given point belongs. Fourth, as an application of the signed barycentric coordinate, the smooth shading of a square with an equilateral triangle inside is performed, along with the generation of repetitive color patterns with the left-right and top-bottom symmetry.
N차원 단체를 형성하는 과도 결정 선형 시스템에서의 여러 유형의 가중 최소 제곱 해에 대한 고찰
김성복 조선대학교 공학기술연구원 2023 공학기술논문지 Vol.16 No.2
This paper makes an in-depth algebraic and geometric analysis on several types of weighted least-squares solutions based on different measures to an overdetermined linear system, in which the number of unknowns is , and the number of constraints is . In this paper, for effective analysis, the barycentric coordinate is adopted to express the position of a given point in -dimensional space with respect to an -dimensional simplex. First, the ordinary least-squares solution and the orthogonal distance least-squares solution are formulated as two approximated solutions to an overdetermined system forming an -dimensional constraint simplex. Next, through algebraic and geometric analysis, the relationship between the ordinary least-squares solution to an overdetermined system and the centroid of the corresponding constraint simplex is identified, the effect of scaling of an overdetermined system on the ordinary least-squares solution is investigated, and the relationship between the orthogonal distance least-squares solution to an overdetermined system and the symmedian point of the corresponding constraint simplex is identified. Finally, simulation results for a two-dimensional overdetermined system forming a constraint triangle are given.