# 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