The Challenge:
Three points are chasing each other:
- Red point (A) chases Blue point (B)
- Blue point (B) chases Green point (C)
- Green point (C) chases Red point (A)
Questions:
- Will these points ever meet?
- If they meet, how long will it take?
- What shape does their path form?
- Can you predict their final meeting point?
And here the solution to our weekly problem no.13!
Solution to the Vector Chase Challenge
Key Findings:
- Yes, the points will always meet eventually!
- They meet at the centroid (center of mass) of the original triangle
- Each point traces a logarithmic spiral path
- The triangle maintains similar shape throughout the chase
Mathematical Properties:
- The center of mass remains fixed throughout the motion
- The triangle shrinks exponentially with time
- The perimeter decreases at a constant rate
- The area decreases quadratically
Mathematical Explanation:
1. Why They Always Meet
- The triangle formed by the three points always remains similar to the original triangle
- The size of the triangle decreases at a constant rate
- This guarantees convergence to a single point
2. Center of Mass
- The center of mass (centroid) remains fixed throughout the motion
- Final meeting point: (xₐ + xᵦ + xᵧ)/3, (yₐ + yᵦ + yᵧ)/3
- This is because the system’s total momentum is conserved
3. Logarithmic Spiral Paths
- Each point traces a logarithmic spiral toward the center
- The angle of pursuit remains constant
- This creates the characteristic spiral pattern
4. Rate of Convergence
- The triangle shrinks exponentially with time
- The rate depends on the initial configuration
- More equilateral initial triangles converge more slowly
Meeting Point Formula:
The points will meet at coordinates:
- X = (x₁ + x₂ + x₃)/3
- Y = (y₁ + y₂ + y₃)/3
Applications:
- Pursuit-evasion games
- Robotic coordination
- Swarm behavior modeling
- Dynamic systems analysis