8.6 Parametric Equations

Parametric Equations

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Parametric to rectangular conversion

Parametric equations define a curve using two separate equations, $x(t)$ and $y(t)$, where $t$ is the parameter that links them together. Instead of writing $y$ directly as a function of $x$, you let both coordinates depend on a third variable. This is especially useful for curves that fail the vertical line test or for describing motion over time.

Converting parametric → rectangular means eliminating the parameter $t$ so you're left with a single equation in $x$ and $y$:

1. Pick whichever parametric equation is easier to solve for $t$
2. Solve that equation for $t$
3. Substitute the result into the other equation

For example, given $x(t) = t^2$ and $y(t) = t^3$:

• Solve for $t$: since $x = t^2$, you get $t = x^{1/2}$ (assuming $t \geq 0$)
• Substitute into $y(t)$: $y = (x^{1/2})^3 = x^{3/2}$

For parametric equations involving trig, use identities instead of solving for $t$ directly. If $x(t) = \cos(t)$ and $y(t) = \sin(t)$, square both and add them: $x^2 + y^2 = \cos^2(t) + \sin^2(t) = 1$. That gives you the rectangular equation of a circle.

Converting rectangular → parametric means introducing a parameter $t$. The simplest approach: let $x(t) = t$, then rewrite $y$ in terms of $t$. For $y = x^2$, you'd write $x(t) = t$ and $y(t) = t^2$. Note that parametric representations aren't unique; you could also use $x(t) = 2t$ and $y(t) = 4t^2$ for the same parabola.

Modeling with parametric equations

Parametric equations are a natural fit for modeling 2D motion, where $x(t)$ and $y(t)$ give the horizontal and vertical position at time $t$.

Projectile motion is the classic example. An object launched with initial speed $v$ at angle $\theta$ from the ground follows:

$x(t) = v\cos(\theta) \cdot t$

$y(t) = v\sin(\theta) \cdot t - \frac{1}{2}gt^2$

Here $g$ is the acceleration due to gravity (approximately $9.8 \text{ m/s}^2$ or $32 \text{ ft/s}^2$). The horizontal component has constant velocity, while the vertical component is affected by gravity, which is why these are easier to handle as two separate equations rather than one rectangular equation.

Modeling curves and shapes is another major use. A cycloid, the path traced by a point on the rim of a rolling circle of radius $r$, has the parametric form:

$x(t) = r(t - \sin(t))$

$y(t) = r(1 - \cos(t))$

This curve would be extremely difficult to express as a single rectangular equation, which is exactly why parametric form is so valuable.

Calculus of parametric curves

The derivatives of $x(t)$ and $y(t)$ with respect to $t$ give you velocity components:

• $\frac{dx}{dt}$ is the horizontal velocity
• $\frac{dy}{dt}$ is the vertical velocity

To find the slope of the tangent line to a parametric curve at a given point, use the chain rule:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

This works as long as $\frac{dx}{dt} \neq 0$ at that point.

The arc length of a parametric curve from $t = a$ to $t = b$ is:

$L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$

Think of this as adding up tiny distances along the curve, where each small piece has a horizontal component $dx$ and a vertical component $dy$.

Geometric interpretation of parameters

What $t$ actually represents depends on the context:

• In motion problems, $t$ is usually time
• In shape problems, $t$ might represent an angle (like for circles and ellipses) or a proportion along the curve

The domain of $t$ controls which portion of the curve you trace out. For the unit circle $x(t) = \cos(t)$, $y(t) = \sin(t)$:

• $0 \leq t \leq 2\pi$ traces exactly one full revolution
• $0 \leq t \leq \pi$ traces only the upper semicircle
• $0 \leq t \leq 4\pi$ traces the circle twice

Changing constants in the parametric equations affects the curve's size and shape. In the cycloid equations, increasing $r$ produces a larger cycloid. For an ellipse written as $x(t) = a\cos(t)$, $y(t) = b\sin(t)$, the values of $a$ and $b$ control the width and height.

Advanced parametric representations

Polar-to-parametric conversion: Any polar equation $r = f(\theta)$ can be written parametrically by using $\theta$ as the parameter:

$x(\theta) = r(\theta)\cos(\theta), \quad y(\theta) = r(\theta)\sin(\theta)$

This is useful when you need to apply rectangular-coordinate techniques (like finding tangent lines) to polar curves.

Parametric surfaces extend the idea to three dimensions. Instead of one parameter, you use two ($u$ and $v$) to define a surface with three equations: $x(u,v)$, $y(u,v)$, and $z(u,v)$. This is a concept you'll encounter more in multivariable calculus, but it's worth knowing that parametric thinking scales beyond 2D.