GROEBNER BASIS SOLUTION OF PLANAR RESECTION PROBLEM
GROEBNER BASIS SOLUTION OF PLANAR RESECTION PROBLEM
复制标题
平面切除问题的 Groebner 基解
DOI：
10.1179/sre.2002.36.285.528
复制
发表时间：
2002
期刊：
影响因子：
1.6
通讯作者：
J. Awange
中科院分区：
文献类型：
--
作者：
J. Awange
摘要 三个非线性平面切除方程通常通过首先分三步求解水平面中的未知方向 σ 来解析求解。第一步是消除步骤，第二步是除法步骤，第三步是再次消除步骤。一旦确定了水平面中的未知方位参数，就将其代入两个初始非线性方程中的任意一个中，以获得观测未知站的位置{x，y}。利用以B. Buchberger算法为计算引擎的Groebner基技术，一步获得未知平面后方交会元素{x, y, σ}（即未知观测站P ε E2的位置{x, y}和未知方位元素σ在水平面中）与已知三个目标点Pi ε E2|ì ε的坐标Xi, Yi|ì ε {1, 2, 3}之间的三个直接关系{1, 2, 3} 以及从观测未知站 P ε E2 到坐标 Xi, Yi|ì ε {I, 2, 3} 已知的已知三个目标点 Pi ε E2|ì ε {I, 2, 3} 进行水平方向观测 Ti|ì ε{I, 2, 3}。这三个直接关系允许更快地计算未知数 {x, y, σ}，并且可以在 sunJeyors、工程师和架构携带到现场的笔记本电脑和可编程科学计算器中轻松实现算法。Groebner 基础方法因此避免了经典分析过程的向前和向后步骤。
Abstract The three nonlinear planar resection equations are usually solved analytically by first solving for the unknown orientation σ in the horizontal plane in a three-step procedure. The .first step is the elimination step, the second step is the division step and the third step is once again elimination step. Once the unknown orientation parameter in the horizontal plane has been determined, it is substituted back in any of the two of the initial nonlinear equations to obtain the position {x, y} of the observing unknown station. By making use of the Groebner basis technique, whose computing engine is the B. Buchberger algorithm, we obtain in a single step three direct relationship between the unknown planar resection elements {x, y, σ} (being the position {x, y} of the unknown observing station P ∈ E2 and the unknown orientation element σ in the horizontal plane) and the coordinates Xi, Yi|ì ∈ {1, 2, 3} of the known three target points Pi ∈ E2|ì ∈ {1, 2, 3} and the horizontal direction observations Ti|ì ∈{I, 2, 3} being made from the observing unknown station P ∈ E2 to the known three target points Pi ∈ E2|ì ∈ {I, 2, 3} whose coordinates Xi, Yi|ì ∈ {I, 2, 3} are known. These three direct relationships allow a faster computation of the unknowns {x, y, σ} and an easy implementation of the algorithm in laptops and programmable scientific calculators being carried by sunJeyors, engineers and architectures to the field The Groebner basis approach thus avoids the forward and backward steps of the classical analytical procedure.