Problem I. 375. (April 2015)
I. 375. In this task we study a mechanical model for the road junction problem by using a computer simulation.
The junction problem asks the following. We are given the location of three towns. Where to put the junction such that the total cost of roads to be built from the junction to the towns is minimal? A following mechanical model for this problem was given by Pólya György (see in his book George Pólya: Induction and Analogy in Mathematics, Princeton Univ. Press, 1954). We fix three pulleys with ropes, join one end of the ropes at a common point, and attach three equal weights on the other ends of the ropes. The equilibrium point of the system (where the forces acting at the common point cancel each other out) gives the location of the road junction.
You should generalize the above model to \(\displaystyle n\) points, and display the resulting system on the screen. To simplify the appearance, we replace each pulley with a hole cut in the plane that passes through the \(\displaystyle n\) points. Each rope is represented by a line segment joining the holes and the common point. The common point cannot be inside a hole. A model for \(\displaystyle n=3\) created by using the Wolfram Language is found at http://demonstrations.wolfram.com/PolyasMechanicalModelForTheFermatPoint/.
Your computer model should have the following properties.
\(\displaystyle \bullet\) The user should be able to set the value of \(\displaystyle n\) between 3 and 8;
\(\displaystyle \bullet\) The user should be able to specify the location of the holes in the plane, but a random hole configuration can also be chosen;
\(\displaystyle \bullet\) Your program approximates the equilibrium state iteratively according to the following rules:
\(\displaystyle a.\) the plane is viewed from above;
\(\displaystyle b.\) forces acting at the common point are represented by unit vectors along the direction of the ropes;
\(\displaystyle c.\) at the next iteration step the common point is moved to the direction determined by the sum of the unit vectors.
You can use any programming environment to solve the task. Your simulation should have a nice appearance and be easy to use.
A description of your solution (i375.pdf) containing the main steps and observations, the source code of your program, and any files necessary to compile or run the program should be submitted in a single compressed file (i375.zip).
Deadline expired on May 11, 2015.
|6 students sent a solution.|
|10 points:||Kovács Balázs Marcell.|
|9 points:||Dombai Tamás, Fényes Balázs.|
|8 points:||2 students.|
|6 points:||1 student.|