Theory

Parametric curve

A parametric curve or parameterized curve in the plane is a function $\gamma :[a,b]\to {\mathbb{R}}^2$ from an interval to the plane, given with coordinate functions by $(x(t),y(t))$.

The data of a parametric curve includes the parametrization, not just the set of points in the image.

Warning - same vertical line

The image of $(x,y)=(a,t^3)$ is the same vertical line!

Warning - Lissajous figure self-intersection

This “Lissajous figure” shows that parametric curves can intersect themselves!

Warning - same image in the upper quadrant

In the upper quadrant, this curve has the same image:
$(x,y)=(e^{-3t},e^{-t})$
Check: $y^3=(e^{-t}{)}^3=e^{-3t}=x.$