Problem description:
There are ants at a certain point A on the surface of a cuboid, along with each vertex of the cuboid. Each ant starts from A and stops when it reaches the top. Requirements:
Ants don't walk inside the cuboid, they can only walk on the surface;
Ants can't stay outside the points on the surface of the cuboid, that is, the points inside and on the surface of the cuboid can be reached by ants.
Q: What is the shortest path for ants to crawl?
Consider expanding a cuboid into a plane figure. The unfolding method is to cut each face along three mutually perpendicular axes and unfold it into a face.
From point A, ants have to go through several faces to reach the vertex of a cuboid.
The points on each extended face are numbered according to different types of faces.
Calculate the coordinates of the point represented by each number, and express the coordinates in three directions with X, Y and Z. ..
Calculate the distance from point A to other points, and then choose the shortest distance. The shortest distance is the shortest path for ants to crawl.
Problem solving skills:
1 projection method
Projection method is a common skill to solve the shortest path problem of rectangular ants. Its basic idea is to expand a cuboid into a plane, and then solve the shortest path on the plane.
The specific steps are as follows: 1. Expanding a cuboid into a plane can be achieved by expanding the faces in a certain order and splicing them together. 2. Mark the starting point and the target point on the plan and connect them. 3. Use the shortest path algorithm in graph theory (such as Dijkstra algorithm or A* algorithm) to calculate the shortest path from the starting point to the target point. 4. By mapping the shortest path back to the surface of the original cuboid, we can get the shortest path for ants to walk on the cuboid.
2 space division method
Spatial segmentation is another skill to solve the shortest path problem of rectangular ants. Its basic idea is to divide a cuboid into several cubes, and then move between cubes to find the shortest path.
The specific steps are as follows: 1. Divide the cuboid into cubes, each cube has six adjacent cubes. 2. Record the shortest path length from the starting point to the current small cube in each small cube. 3. Use dynamic programming or breadth-first search algorithm to update the shortest path length in each small cube step by step until the target point is reached. 4. According to the recorded shortest path length, the shortest path of ants walking on a cuboid can be obtained by tracing the path of ants in reverse.
3 Mathematical modeling method
Mathematical modeling is an abstract mathematical problem-solving skill. It describes the shortest path problem of cuboid ants based on mathematical models and equations, and obtains the optimal solution by solving these equations.
The specific steps are as follows: 1. The starting point and the target point are represented as points in the coordinate system. 2. Establish a mathematical model to describe the walking rules and constraints of the cuboid surface. 3. According to this model, a set of equations is established to represent the problem. 4. Solve this set of equations by numerical calculation method (such as iterative method or optimization algorithm) and get the optimal solution. ?