7.1 Parametric Equations

Parametric equations offer a powerful way to describe curves using a parameter. They allow us to express complex shapes and motions that might be difficult to represent with standard functions. This approach opens up new possibilities for analyzing and visualizing mathematical relationships.

By using parametric equations, we can plot curves, convert between different forms, and represent basic shapes like lines and circles. We can also dive into more advanced concepts like cycloids and motion analysis, giving us tools to tackle real-world problems in physics and engineering.

Parametric Equations

Plotting parametric curves

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• Parametric equations express coordinates of curve points using independent parameter $t$
  • x-coordinate defined by function $x(t)$
  • y-coordinate defined by function $y(t)$
• Plotting curve from parametric equations involves:
  1. Selecting range of $t$ values
  2. Evaluating $x(t)$ and $y(t)$ for each $t$ to obtain $(x, y)$ coordinates
  3. Plotting points on coordinate plane and connecting to form curve
• Examples:
  • Plotting circle with parametric equations $x = \cos(t)$, $y = \sin(t)$, $0 \leq t \leq 2\pi$
  • Plotting spiral with equations $x = t\cos(t)$, $y = t\sin(t)$, $0 \leq t \leq 4\pi$

Conversion to rectangular form

• Converting parametric equations to rectangular form eliminates parameter $t$ yielding single equation in $x$ and $y$
• Parameter elimination methods include:
  • Solving one equation for $t$ and substituting into other equation
  • Using trigonometric identities for equations with trigonometric functions
  • Algebraically manipulating to isolate $y$ in terms of $x$
• Resulting $x$ and $y$ equation represents curve's rectangular form
• Example: Converting $x = 2\cos(t)$, $y = 3\sin(t)$ to $\frac{x^2}{4} + \frac{y^2}{9} = 1$ (ellipse)

Parametric equations for basic shapes

• Lines:
  • Parametric line equations: $x = a + bt$, $y = c + dt$
    • $(a, c)$ represents point on line
    • $(b, d)$ represents vector parallel to line
  • Line slope given by $\frac{d}{b}$
  • Example: $x = 1 + 2t$, $y = 3 + 4t$ represents line with slope $2$ passing through $(1, 3)$
• Circles:
  • Parametric circle equations with center $(h, k)$ and radius $r$: $x = h + r\cos(t)$, $y = k + r\sin(t)$, $0 \leq t \leq 2\pi$
    • Parameter $t$ represents angle from positive x-axis to radius vector (polar coordinates)
  • Example: $x = 2 + 3\cos(t)$, $y = 1 + 3\sin(t)$ represents circle with center $(2, 1)$ and radius $3$

Interpretation of cycloid equations

• Cycloid: curve traced by point on circumference of rolling circle along straight line without slipping
• Parametric cycloid equations:
  • $x = r(t - \sin(t))$
  • $y = r(1 - \cos(t))$
    • $r$: radius of rolling circle
    • $t$: angle of circle rotation
• Cycloid properties:
  • Periodic curve with period $2\pi$
  • x-coordinate advances by $2\pi r$ per complete circle rotation
  • y-coordinate oscillates between $0$ and $2r$
• Cycloid applications:
  • Brachistochrone curve (fastest descent path between two points) is inverted cycloid
  • Cycloid curves used in gear design and rolling motion analysis
• Example: For cycloid with $r = 2$, point traces path as circle rolls, with x-coordinate advancing by $4\pi$ per rotation and y-coordinate varying between $0$ and $4$

Motion Analysis with Parametric Equations

• Velocity vector: Describes the instantaneous rate of change of position
  • Given by $\vec{v}(t) = \langle x'(t), y'(t) \rangle$
• Acceleration vector: Represents the rate of change of velocity
  • Calculated as $\vec{a}(t) = \langle x''(t), y''(t) \rangle$
• Tangent line: Represents the instantaneous direction of motion
  • Slope at point $(x(t), y(t))$ given by $\frac{dy/dt}{dx/dt}$