# Curves defined by parametric equations

## Definition and graphs

- Suppose given two functions \(f(t)\) and \(g(t)\) with a common domain \(I\).
- How we will want to interpret this situation is to consider the corresponding
*parametric equations* \[
x=f(t),\quad y=g(t).
\]
- That is, just as up to now we've been investigating number-valued functions, which assign to \(t\) the number \(f(t)\), now to each \(t\) we assign the plane point \((x,y)=(f(t),g(t))\).
- In this context, \(t\) is called a
*parameter*.
- The collection of the points \((f(t),g(t))\) for \(t\) in \(I\) is a plane curve \(C\). It is called a
*parametric curve*. If no domain \(I\) is given, then we make it the intersection of the domains of \(f\) and \(g\), that is the collection of \(t\) such that both \(f(t)\) and \(g(t)\) makes sense.

- Example. Consider the parametric equations \[
x=t-1,\quad y=t^2+1.
\]
- In case at least one of the functions is linear, we can reparametrize the curve as a graph.
- Substituting \(t=x+1\), we get the graph \[
y=(x+1)^2+1=x^2+2x+2.
\]

- Example. Consider the parametric equations \[
x=e^{2t}-1,\quad y=e^t+1.
\]
- In this case, we can see that both functions have the same function \(u(t)=e^t\) substituted into them. Since the domain for \(t\) is \(-\infty<t<\infty\), the domain for \(u\) is \(0<u<\infty\). Therefore, we get \[
x=u^2-1,\quad y=u+1,\quad0<u<\infty,
\] from which via \(u=y-1\) we get \[
x=y^2-2y,\quad1<y<\infty.
\]

- Exercises. 10.1. 5, 9, 14, 15, 18

## Circles.

- Example. Consider the parametric equations \[
x=\cos\theta,\quad y=\sin\theta,\quad0\le\theta\le2\pi.
\]
- They give as parametric curve the unit circle.
- Note that this is not a graph, since there are both vertical or horizontal lines, which it intersects more than once.
- Note also that by restricting the domain to \(0\le\theta\le\pi\), we get the graph \[
y=\sqrt{1-x^2},\quad-1\le x\le1.
\]

- In general, the standard parametrization of the circle with centre \((h,k)\) and radius \(r\) is \[
x=r\cos\theta+h,\quad y=r\sin\theta+k.
\]
- Exercises. 10.1. 11, 13, 19, 21