parametric equation
Parametric equation
Parametric equations define a set of quantities (like \[x\] and \[y\]) using a third variable, known as a parameter (sometimes denoted as \[t\] or \[\theta\]). Instead of directly relating \[y\] to \[x\] (e.g. \[y=x^{2}\]), \[x\] and \[y\] are expressed as separate functions of the parameter.
We use parametric equations when ordinary functions fall short. A common example would be to graph a unit circle. The equation for a unit circle is \[x^{2}+y^{2}=1\]. However, we can't rewrite this into a single function as \[y=\pm \sqrt{1-x^{2}}\], which very obviously will fail the vertical line test. Therefore, we define \[\left( x,y \right)\] as \[\left( \cos(t),\sin(t) \right)\], or in equation form, \[x=\cos(t),y=\sin(t)\] (note that we can also write it as \[x(t)=\cos(t)\]). Now one parameter, \[t\] corresponds to two outputs \[x\] and \[y\], instead of being an ordinary function, where an input \[x\] corresponds to only one output \[y\].
The main difference between ordinary functions and parametric equations is that the restriction for a single \[y\] value for each \[x\] is gone, which makes it much easier to describe curves like circles and loops without having to break them into multiple equations. Another use case for parametric equations would be to describe motion, e.g. \[x(t)=r\cos \left( \omega t \right),y(t)=r\sin \left( \omega t \right)\].
Parametric equation are convenient for describing curves in higher-dimensional spaces, e.g. a three-dimensional helix: \[\mathbf{r}(t)=\left( x(t),y(t),z(t) \right)=\left( a\cos(t),a\sin(t),bt \right)\].
Implicitization
Converting a set of parametric equations to a single implicit equation (a single function of several variables) involves eliminating the variable \[t\] from the equations \[x=f(t),y=g(t)\]. Solving \[y=g(t)\implies t=g^{-1}(y)\] and using this in \[x=f(t)\] gives the equation \[x=f(g^{-1}(y))\].
To retake the example of a circle of radius \[a\], the parametric equations would be \[x=a\cos(t),y=a\sin(t)\]. Then it can be rewritten as \[\frac{x}{a}=\cos(t),\frac{y}{a}=\sin(t)\],
Another thing to keep in mind is that after we eliminate the parameter variable (\[t\]), the resulting equation will hold true for every point in the parametric graph, while additional points may also satisfy it.
A classic example would be to define \[x=t^{2},y=t^{2}+1\]. After eliminating \[t\], we get \[y=x+1\], which holds true for every \[t\]. However, for the original parametric equation, there is an additional hidden constraint being \[x,y\ge0\] (as \[t^{2}\ge0\]), while \[y=x+1\] does not have this constraint. Therefore, \[y=x+1\] would also hold true for additional points, e.g. when \[y=-5,x=-4\].