8.6 Parametric Equations

Parametric equations offer a powerful way to describe curves and motion using a parameter. They allow us to represent complex shapes and trajectories that can't be easily expressed with standard functions. This approach opens up new possibilities for modeling and analysis.

Converting between parametric and rectangular forms is crucial for working with these equations. We can also use calculus techniques to find tangent lines, velocities, and arc lengths. Understanding how the parameter affects curve shape and position is key to mastering this topic.

Parametric Equations

Parametric to rectangular conversion

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• Parametric equations define a curve using two equations $x(t)$ and $y(t)$ with parameter $t$ (circle)
• Convert parametric to rectangular by eliminating $t$
  • Solve one equation for $t$ and substitute into the other
  • $x(t) = t^2$, $y(t) = t^3$, solve $x(t)$ for $t = \sqrt{x}$, substitute into $y(t)$ to get $y = x^{3/2}$
• Convert rectangular to parametric by introducing $t$
  • Express $x$ and $y$ as functions of $t$
  • $y = x^2$, let $x(t) = t$ and $y(t) = t^2$

Modeling with parametric equations

• Model 2D motion with $x(t)$ and $y(t)$ representing position at time $t$
  • Projectile launched with velocity $v$ at angle $\theta$: $x(t) = v\cos(\theta)t$ and $y(t) = v\sin(\theta)t - \frac{1}{2}gt^2$, $g$ is gravity
• Model shapes and curves
  • Cycloid (path traced by point on rolling circle): $x(t) = r(t - \sin(t))$ and $y(t) = r(1 - \cos(t))$, $r$ is circle radius
• Vector-valued functions can represent parametric curves in higher dimensions

Calculus of parametric curves

• Derivatives provide curve information
  • First derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are velocity components
  • Second derivatives $\frac{d^2x}{dt^2}$ and $\frac{d^2y}{dt^2}$ are acceleration components
• Find tangent line slope using first derivatives: $\frac{dy/dt}{dx/dt}$
• Calculate arc length with integrals: $\int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$

Geometric interpretation of parameters

• Parameter $t$ represents different quantities based on context
  • Motion problems: $t$ is often time
  • Shape problems: $t$ may be angle or proportion
• Parameter domain determines portion of curve considered
  • Unit circle $x(t) = \cos(t)$, $y(t) = \sin(t)$, domain $0 \leq t \leq 2\pi$ is one revolution
• Parameter values affect curve shape and position
  • Cycloid equations $x(t) = r(t - \sin(t))$, $y(t) = r(1 - \cos(t))$, changing $r$ changes cycloid size

Advanced parametric representations

• Polar coordinates can be expressed parametrically as $x(t) = r(t)\cos(t)$, $y(t) = r(t)\sin(t)$
• Phase plane analysis uses parametric equations to study dynamical systems
• Parametric surfaces extend the concept to three dimensions, defining a surface with three equations $x(u,v)$, $y(u,v)$, and $z(u,v)$