삼각형에서의 부호를 갖는 질량 중심 좌표에 대한 체계적 분석과 사각형 스무스 컬러 세이딩에의 활용

Abstract

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.