Intersection and Collision in 3D

Key Concepts

Intersection vs. Collision:
- Intersection: Paths cross at the same point but possibly at different times.
- Collision: Paths cross at the same point and the same time.
Graphical vs. Algebraic Approach:
- Graphical representation is not effective in 3D.
- Algebraic/symbolic methods are necessary to determine intersections or collisions.

Objective: Determine if two trajectories, (r(t)) and (u(t)), intersect or collide.
Variables Used:
- (t) for the trajectory (r)
- (s) for the trajectory (u)

Setup Equations:
- Analyze each coordinate (x, y, z) separately.
- Set equations for each coordinate assuming they have the same values at (t) for (r(t)) and (s) for (u(s)).
- Example equations:
  - X-coordinate: (t = 1 + 2s)
  - Y-coordinate: (t = 1 + 6s)
  - Z-coordinate: (t^3 = 1 + 14s)
Solve Equations:
- Start with the simplest equation (usually x or y) due to linearity.
- Solve for (t) and (s) values.
- Substitute into other equations to check consistency.
Determine Possible Solutions:
- Solutions for (s) and (t):
  - (s = 0, t = 1)
  - (s = \frac{1}{2}, t = 2)
- These solutions satisfy x and y coordinates.
Check Z-coordinate Consistency:
- Ensure the z-values are equal for the solved (s) and (t).
- Check both sets of values:
  - (s = 0, t = 1): Z-values match.
  - (s = \frac{1}{2}, t = 2): Z-values match.

Intersections Found:
- Intersection points:
  - ((1, 1, 1))
  - ((2, 4, 8))
Collision Check:
- No collisions since no common (s) and (t) values that satisfy all three coordinate equations simultaneously.

Understanding the conditions for intersections and collisions through equations is crucial.
Visualization helps in understanding the concept but algebra confirms the behavior.
Important to distinguish between intersection and collision with respect to time.