Lecture Notes: Parametric Equations and Cycloids

Review of Intersection of Planes

• Last Lecture: Discussed 3x3 linear systems as intersections of planes.
  • Unique solution: intersection at a single point.
  • No solutions or infinitely many solutions: planes are parallel or coincide.

Parametric Equations of Lines

• Line Representation: Intersection of two planes or using parametric equations.
• Parametric Equation: Describes the trajectory of a moving point along a line.
• Example:
  • Given two points on a line: (-1,2,2) and (1,3,-1).
  • Find equation of line passing through these points.
  • Derive coordinates of any point on the line as functions of a parameter t.
• Equation Derivation:
  • Let Q(t) be a moving point.
  • Parametrize the line as Q(0) = (-1,2,2) and Q(1) = (1,3,-1).
  • Find coordinates using linear interpolation:
    • x(t) = -1 + 2t
    • y(t) = 2 + t
    • z(t) = 2 - 3t

Line's Intersection with a Plane

• Example Analysis: Line intersects plane x + 2y + 4z = 7.
• Check Relative Positions:
  • Evaluate plane equation at given points.
  • Determine if points lie on the same side, opposite sides, or one on the plane.
• Plugging parametric equations: Solve to find t where line intersects the plane (t = 1/2).
• Resulting Point: Q(1/2) = (0, 5/2, 1/2).
• Special cases: Line fully contained or parallel to the plane (no solution or consistent results).

General Parametric Equations

• Trajectories of Points: Extend to arbitrary motions in 2D or 3D space.
• Cycloid Curve:
  • A point on the rim of a rolling wheel (radius a).
  • Parametrized by angle θ (wheel rotates).
  • Use trigonometry to find parametric equations.
• Vector Breakdown:
  • Decompose into simpler vectors for easier understanding: OA, AB, BP.
  • Derive components:
    • OA = (aθ, 0)
    • AB = (0, a)
    • BP = (-a sin θ, -a cos θ)
  • Combine for final parametric equations:
    • x(θ) = aθ - a sin(θ)
    • y(θ) = a - a cos(θ)

Analysis Near Critical Points

• Taylor Expansion: Approximate sine and cosine for small angles.
• Cycloidal Motion: Analyze behavior near the bottom of the curve.
  • Find y ≈ (θ^2)/2 and x ≈ (θ^3)/6 for small θ.
  • Conclude: Near the bottom, the motion has a vertical tangent (infinite slope).

Conclusion

• Application: Study cycloidal motion to understand parametric representations better.
• Next Class: Practice exams will be provided for the upcoming test.