Lecture: Sketching Parametric Curves

Introduction

• Discussion on sketching parametric curves
• Parametric curves as trajectories evolving over time
• Also known as vector-valued functions

Understanding Vector-Valued Functions

• A new way to look at functions
• Uses formulas, graphs, and tables of values
• Example: Visiting range of values for t, generating corresponding x, y values

Example: Quadratic Nature of Vector Functions

• x = t^2, y = t
• Analyzing with a table for t: -2 to 2:
  • x: 4, 1, 0, 1, 4
  • y: -2, -1, 0, 1, 2
• Plotting these values shows a sideways parabola
  • Curve implies a direction of particle movement

Graphing Considerations

• Graphing technique: infer continuity between points
• Parabola traced out shows directionality
• Curve is infinitely long; practical bounds are set

Second Example: Circular Trajectory

• Function: v(t) = (4cos(t), -2sin(t))
• Expectation: Circular trajectory transformed into an ellipse
• Table of values for t from 0 to 2π:
  • x: 4, ~2.8, 0, -4
  • y: 0, ~-1.4, -2, 0
• Pattern: Elliptical with axes 4 and 2 long
• Directionality indicated by negative sine; clockwise motion

Domain Considerations

• New functions: Involving square root and reciprocal functions
• Domain restrictions: Valid for t > 0 for both square root and 1/t

Example with Domain Restrictions

• x = √t, y = 1/t
• Example values: t = 0.1, 0.2, 0.5, 1, 2, 4, 9
  • x: ~0.3, ~0.45, ~0.7, 1, √2, 2, 3
  • y: 10, 5, 2, 1, 0.5, 0.25, ~0.11
• Trajectory shows fast motion that slows down over time

Animation Insights

• Animation reveals motion speed and trajectory
• Fast initial motion, followed by slower motion
• Importance of visualizing motion to understand speed, velocity, and acceleration

Conclusion

• Parametric curve sketching offers insights into particle trajectories
• Graphing and animations help reveal underlying physical concepts