11.1 Parametric Equations and Curves

Parametric equations let us describe curves using a third variable, t. This opens up new ways to represent and analyze shapes in the xy-plane, going beyond traditional Cartesian coordinates.

In this section, we'll learn how to define, manipulate, and apply parametric equations. We'll explore their uses in modeling real-world phenomena like cycloids, projectile motion, and particle trajectories.

Parametric Equations and Curves

Defining Parametric Equations and Curves

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• Parametric equations represent a curve in the $xy$-plane using two equations, $x=f(t)$ and $y=g(t)$, where $t$ is the parameter
  • The parameter $t$ is an independent variable that determines the position of a point on the curve
  • As $t$ varies, the point $(x,y)=(f(t),g(t))$ traces out a parametric curve in the plane
• A parametric curve is a curve in the plane that is defined by parametric equations
  • The curve is the path traced out by the point $(x,y)$ as the parameter $t$ varies over a given interval
  • Examples of parametric curves include circles, ellipses, and cycloids

Manipulating Parametric Equations

• Eliminating the parameter involves solving the parametric equations for $t$ and substituting one equation into the other to obtain a Cartesian equation in $x$ and $y$
  • This process can be used to convert parametric equations into a more familiar form
  • For example, the parametric equations $x=\cos t$ and $y=\sin t$, $0 \leq t \leq 2\pi$, represent a circle with radius 1 centered at the origin
    • Squaring and adding the equations yields $x^2+y^2=1$, the Cartesian equation of the circle
• Parametric equations can be manipulated algebraically, such as adding, subtracting, or multiplying the equations, to transform the curve or simplify the equations
  • These manipulations can help reveal properties of the curve or make it easier to graph

Applications of Parametric Equations

Modeling Cycloids

• A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping
  • Parametric equations for a cycloid are $x=a(t-\sin t)$ and $y=a(1-\cos t)$, where $a$ is the radius of the wheel
  • Cycloids have applications in physics, such as modeling the path of a pendulum or the shape of a rope hanging under its own weight (catenary)

Describing Projectile Motion

• Projectile motion can be modeled using parametric equations, where $x=v_0t\cos\theta$ and $y=v_0t\sin\theta-\frac{1}{2}gt^2$
  • $v_0$ is the initial velocity, $\theta$ is the angle of projection, $g$ is the acceleration due to gravity, and $t$ is time
  • These equations describe the path of a projectile, such as a ball thrown or a bullet fired, assuming no air resistance
  • The equations can be used to determine the range, height, and time of flight of the projectile

Analyzing Particle Motion

• Parametric equations are useful for describing the motion of a particle in the plane
  • The position of the particle at time $t$ is given by the coordinates $(x(t),y(t))$
  • The velocity and acceleration of the particle can be found by differentiating the position equations with respect to $t$
• Parametric equations can model various types of particle motion, such as circular motion, harmonic motion, or more complex paths
  • For example, the equations $x=\cos(2t)$ and $y=\sin(3t)$ describe a particle moving in a figure-eight pattern, known as a Lissajous curve