Construction of line Layout Model Based on Geometric Deduction and Constrained Heuristic Search

Abstract

Based on problem B of the National Mathematical Contest in Modeling for College Students in 2023, this paper establishes a model through geometric deductive deduction, and establishes and solves an optimal model of survey line layout through constrained heuristic algorithm. In problem 1, a mathematical model of coverage width and overlap rate is established based on triangle relation and triangle sine theorem by finding the mathematical relationship between target quantity and known quantity. In problem 2, the angle between the known direction of the survey line and the projection of the normal direction of the submarine slope on the horizontal plane is . We can imagine a surface perpendicular to the direction of the measurement line, and then find the slope of the imaginary surface according to the geometric relationship, and replace the slope Angle of problem 1 with the imaginary slope Angle to obtain the mathematical model of the coverage width. Based on this, the constraints of the Angle and overlap rate of the line are added, and the problem 3 is simplified into the optimization problem of multiple routes in problem 2.  Finally, by changing the parameter setting of slope Angle in the model, the sensitivity analysis is carried out, and it is found that the model is stable and reliable.To sum up, the problems from one to four are gradually deepened and progressively advanced, and the development of Marine survey field is facilitated by establishing a model of depth survey line layout.