# Parametric Curves A **parametric equation** defines a group of quantities as functions of one of more independent variables called parameters. For example, the equations below form a parametric representation of the unit circle, where $t$ is the parameter: $x = cos(t)$ $y = sin(t)$ [Wikipedia has some great examples](https://en.wikipedia.org/wiki/Parametric_equation#Examples_in_two_dimensions). Note that parametric curves are a great example of [finding different ways to represent the same underlying concept](Big%20Idea%20Representation.md). Note that in the above example, we can think of our parametric equation as really being a function that maps from $\mathbb{R}$ to $\mathbb{R}^2$: $f: \mathbb{R} \rightarrow \mathbb{R}^2$ Written in vector notation: $f(t) = \begin{bmatrix} cos(t) \\ sin(t) \\ \end{bmatrix}$ A concrete example of this type of view is seen [here](https://www.youtube.com/watch?v=bb4bSCjlFAw&list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7&index=6) with 3b1b. #### Key idea When thing to keep in mind when using parametric curves is that you aren't quite sure (based on simply the visualization of the curve) what input value(s) map to a particular output. You simply have the output curve. In a sense what is so interesting about parametric curves is that (in many examples at least) you are going from 1 to 2 dimensions. This clearly means that the functions themselves provide a great deal of information in the form of *constraints*, since we can't just create information out of thin air. The information is clearly stored in the function. If we just want some analytical way of describing curves, you just find a parametric function that does it, and you don't care about the rate. An example is [here](https://youtu.be/bb4bSCjlFAw?list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7&t=287). What this example shows is that there are *two* different functions that produce the exact same **parameterize**. > Functions will *parameterize* a curve if, when you draw just in the output space, you get that curve. #### Another key idea See this [fantastic answer here](https://math.stackexchange.com/questions/485407/what-is-a-the-intuition-behind-a-parametric-equation) on SE. The key takeaway is that we are able to introduce a time variable that allows us to treat out $x$ and $y$ output variables independently. So, given time (our *parameter*), we no longer need to explicitly follow the relationship of $x$ and $y$. --- Date: 20210524 Links to: [Mathematics MOC](Mathematics%20MOC.md) [DDG-Overview](DDG-Overview.md) References: * [Parametric vs. Cartesian, black pen red pen](https://www.youtube.com/watch?v=EliQoqfT9GE) * [wikipedia](https://en.wikipedia.org/wiki/Parametric_equation) * My handwritten notes, high dimensional probability theory * [Parametric curves, khan/3b1b](https://www.youtube.com/watch?v=bb4bSCjlFAw&list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7&index=6) * [Parametric surfaces, khan/3b1b](https://youtu.be/345SnWfahhY?list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7)