Curves defined by parametric equations

Introduction

Graphs of the form $y=f(x)$ are powerful, but they cannot represent a closed loop, a vertical tangent, or a curve that folds back on itself. Parametric equations lift that restriction by letting both coordinates move independently with a third variable, usually written $t$.

Parametric form

A plane curve can be given by

As $t$ runs through an interval $I$, the point $(x(t),y(t))$ traces the curve. Different parameterizations can describe the same geometric object — think of walking the same path at different speeds.

The cycloid

Place a circle of radius $r$ on the $x$-axis and roll it to the right without slipping. A marked point on the rim starts at the origin. After the circle has rotated through an angle $\theta$, the center has moved a distance $r\theta$, and the marked point is at

One full arch corresponds to $\theta$ running from $0$ to $2\pi$. The curve meets the axis again at $(2\pi r,0)$.

Interactive cycloid

Watch the generating circle roll and the marked point leave the cycloid behind.

Teaching video

A short lecture on the rolling constraint and why the cycloid is the right first example in Calculus III.

Arc length

Area under one arch

Takeaways