8.4: Plane Curves and Parametric Equations

Consider a curve in the plane. When drawing it, the tip of your pencil is at a point for one moment in time.

Insight: This point is moving through the plane with respect to time. We can represent this point moving behavior with plane curves.

If $f$ and $g$ are functions defined on an interval $I$, then the set of all points $(f(t), g(t))$ is called a plane curve. The equations \[x = f(t) \qquad y = g(t)\] are called parametric equations with parameter $t$.

Sketch the plane curve defined by the parametric equations \[x = \cos t \qquad y = \sin t\] Remember to indicate which direction the point is moving on the curve.

Sketch the plane curve defined by the parametric equations \[x = t^2 - 3t \qquad y = t - 1\]

Sketch the plane curve defined by the parametric equations \[x = 2t \qquad y = t + 6\]

Eliminating the Parameter

If we eliminate the parameter $t$, then we have a rectangular equation.

Eliminate the parameter for the parametric equations \[x = t^2 - 3t \qquad y = t - 1\]

Eliminate the parameter for the parametric equations \[x = \cos t \qquad y = \sin t\]

Polar Equations in Parametric Form

Consider a polar equation $r = f(\theta)$. Recalling that \[x = r\cos \theta \qquad y = r \sin \theta\] we replace $r$ with $f(\theta)$ to get \[x = f(\theta)\cos \theta \qquad y = f(\theta) \sin \theta\] But these are parametric equations! Instead of using $\theta$, which represents angle, we use a general parameter $t$:

The graph of the polar equation $r = f(\theta)$ is the same as the plane curve with parametric equations \[x = f(t) \cos t \qquad y = f(t) \sin t\]

Consider the equation $r = \theta$. Express this equation in parametric form. What is the graph?

Consider the equation $r = 1 + \cos \theta$. Express this equation in parametric form.

In Section 8.2, we saw the length $r$ as an "arm" tracing out the polar graph.

I use these set of parametric equations to draw this arm!