10.6 Parametric Equations

Parametric equations offer a powerful way to describe curves and motion in mathematics. They use a parameter, often time, to define x and y coordinates separately, allowing for more flexible representations of complex shapes and movements.

These equations shine in modeling real-world scenarios like projectile motion and planetary orbits. They're also crucial in computer graphics and animation. By mastering parametrics, you'll gain valuable tools for analyzing motion and solving advanced mathematical problems.

Parametric Equations

Parametric vs rectangular equations

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• Parametric equations define a curve using a parameter $[t](https://www.fiveableKeyTerm:t)$ (time, angle, or other variable) while rectangular equations directly relate $x$ and $y$ coordinates
• Parametric form: $x = f(t)$ and $y = g(t)$ where $f(t)$ and $g(t)$ are functions of $t$ (sine, cosine, polynomial)
• Rectangular form: $y = f(x)$ or $F(x, y) = 0$ expressing $y$ directly in terms of $x$ or as an implicit function
• Some curves (circles, ellipses, cycloids) are easier to represent using parametric equations

Graphing parametric curves

• Create a table of $t$, $x$, and $y$ values by substituting $t$ values into the parametric equations
• Plot points $(x, y)$ on the coordinate plane and connect them in order of increasing $t$ to form the curve
• Curve direction determined by increasing $t$ values (counterclockwise for increasing angles)
• Find $x$ and $y$ intercepts by setting $y = 0$ and $x = 0$ respectively and solving for $t$
• Identify any asymptotes, cusps, or self-intersections by analyzing the behavior of the parametric equations

Applications of parametric equations

• Model real-world situations involving motion, curves, and surfaces (projectile motion, planetary orbits, Lissajous figures)
• Describe position of an object moving in a plane with $x(t)$ and $y(t)$ representing coordinates at time $t$
• Formulate parametric equations based on given information (initial position, velocity, acceleration) to answer questions about the situation
• Parametric equations used in computer graphics, animation, and CAD software to generate curves and surfaces
• Calculate arc length of a parametric curve using the formula: $L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$

Motion analysis with parametrics

• Determine velocity and speed using derivatives:
  1. $x$-velocity: $v_x = \frac{dx}{dt}$
  2. $y$-velocity: $v_y = \frac{dy}{dt}$
  3. Speed: $\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$
• Find acceleration using second derivatives:
  1. $x$-acceleration: $a_x = \frac{d^2x}{dt^2}$
  2. $y$-acceleration: $a_y = \frac{d^2y}{dt^2}$
• Analyze motion at a specific time by evaluating position, velocity, and acceleration equations
• Determine tangential and normal components of acceleration:
  • Tangential (along the curve): $a_T = \frac{d}{dt}(\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2})$
  • Normal (perpendicular to the curve): $a_N = \frac{|v_x a_y - v_y a_x|}{\sqrt{v_x^2 + v_y^2}}$

Advanced topics in parametric equations

• Vector-valued functions: Represent parametric curves in higher dimensions using vectors
• Parametric surfaces: Extend parametric equations to three dimensions to describe surfaces in space
• Phase plane: Use parametric equations to analyze the behavior of dynamical systems
• Differential equations: Solve and interpret parametric equations arising from differential equations