Determine if chasing points will meet
Problem Description
You are given N points in a 2D plane, indexed from 0 to N-1, which form a closed cycle.
At time t = 0, each point i is located at initial coordinates (xi, yi).
All points move simultaneously and continuously. Each point maintains a constant speed v = 1. At any given time t, point i moves directly towards the current position of point (i+1) mod N (i.e., it "chases" the next point in the cycle).
Task
Your program must determine whether all N points will converge and meet at the exact same location at the exact same time.
- If the points do meet, output the meeting time
Tand the meeting position(X, Y). - If the points do not meet, output the string
NEVER.
Input Format
- An integer
N, where2 ≤ N ≤ 200. - Following
Nlines, each containing two real numbersxi yirepresenting the initial coordinates of pointi.
Output Format
- Either the string
NEVER, or three real numbersT X Y.
Requirements and Notes
- All numeric outputs must be within
1e-4absolute error. - Numerical simulation (time-stepping) is an acceptable approach, provided it meets the specified error requirement and runs efficiently.
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Balancing the need for high numerical precision and stability in a continuous, interdependent system (differential equations) against computational efficiency, or recognizing a key geometric invariant for an analytical solution.
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Determine if chasing points will meet
Problem Description
You are given N points in a 2D plane, indexed from 0 to N-1, which form a closed cycle.
At time t = 0, each point i is located at initial coordinates (xi, yi).
All points move simultaneously and continuously. Each point maintains a constant speed v = 1. At any given time t, point i moves directly towards the current position of point (i+1) mod N (i.e., it "chases" the next point in the cycle).
Task
Your program must determine whether all N points will converge and meet at the exact same location at the exact same time.
- If the points do meet, output the meeting time
Tand the meeting position(X, Y). - If the points do not meet, output the string
NEVER.
Input Format
- An integer
N, where2 ≤ N ≤ 200. - Following
Nlines, each containing two real numbersxi yirepresenting the initial coordinates of pointi.
Output Format
- Either the string
NEVER, or three real numbersT X Y.
Requirements and Notes
- All numeric outputs must be within
1e-4absolute error. - Numerical simulation (time-stepping) is an acceptable approach, provided it meets the specified error requirement and runs efficiently.