Parametric Equations:another process to describe a plane curve on the Cartesian coordinate system.

11.1 – Parametrizations of Plane Curves

The coordinates (x and y) are described by the same auxiliary variable called the parameter (t).

There are different parameterizations that produce the same graph or curve.

Parametric equations are continuous on an interval.

The parameter (t) usually represent time. As it increases, it gives direction to the curve.

11.1 – Parametrizations of Plane Curves

Simple curve – distinct values of the parameter yield distinct points. No overlapping occurs.

Closed curve – initial and final values of the parameter on an interval yield the same point.

A simple curve can be drawn without lifting the pencil from the paper, and without passing through any point twice.

A closed curve has the same starting and ending points, and is also drawn without lifting the pencil from the paper.

Simple; closed

Simple; not closed

Not simple; closed

Not simple; not closed

11.1 – Parametrizations of Plane Curves

𝑥=2 𝑡−1 𝑦=𝑡+10≤ 𝑡≤2

𝑥=2 𝑡3+1 𝑦=𝑡 3+2−1≤𝑡 ≤1

𝑥=3√𝑡−3 𝑦=2√4− 𝑡 3≤𝑡 ≤4

𝑥=3 sin (𝑡)𝑦=−cos (𝑡 )0 ≤𝑡≤2𝑟

11.1 – Parametrizations of Plane Curves