# Parametric Equations

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## Parametrized Curves

We often graph functions by letting one coordinate be dependent on another. For example, graphing the function $f(x) = y = x^2$ has y values that we trace out depend upon x values. However, it is very useful to graph functions by letting each coordinate be equal to an equation of an independent variable, known as a parameter. Changing the value of the parameter can change value of any coordinate being used. We choose a range of values for the parameter, and the values that our function takes on as the parameter varies traces out a curve, known as a parametrized curve.

### Parametrized Circle

One curve that can be easily parametrized is a circle of radius one:

We use the variable t as our parameter, and x and y as our normal Cartesian coordinates.

We now let $x = cos(t)$ and $y = sin(t)$, and let t take on all values from $0$ to $2\pi$.

When $t=0$, the coordinate $(1,0)$ is hit. As t increases, a circle is traced out as x initially decreases, since it is equal to the cosine of t, and y initially increases, since y is equal to the sine of t. The circle continues to be traced until t reaches $2\pi$, which gives the coordinate $(1,0)$ once again.

It is also useful to write parametrized curves in vector notation, using a coordinate vector: $\begin{bmatrix} x \\ y\\ \end{bmatrix}= \begin{bmatrix} cos(t) \\ sin(t)\\ \end{bmatrix}$

Click to see a circle drawn parametrically:

### Butterfly Curve

Parametric construction of the butterfly curve

Sometimes curves which would be very difficult or even impossible to graph in terms of elementary functions of x and y can be graphed using a parameter. One example is the butterfly curve.

This curve uses the following parametrization:

$\begin{bmatrix} x \\ y\\ \end{bmatrix}= \begin{bmatrix} \sin(t) \left(e^{\cos(t)} - 2\cos(4t) - \sin^5\left({t \over 12}\right)\right) \\ \cos(t) \left(e^{\cos(t)} - 2\cos(4t) - \sin^5\left({t \over 12}\right)\right)\\ \end{bmatrix}$

## Parametrized Surfaces and Manifolds

In the above cases only one independent variable was used, creating a parametrized curve. We can use more than one independent variable to create other graphs, including graphs of surfaces. For example, using parameters s and t, the surface of a sphere can be parametrized as follows: $\begin{bmatrix} x \\ y\\ z\\ \end{bmatrix}= \begin{bmatrix} sin(t)cos(s) \\ sin(t)sin(s) \\cos(t) \end{bmatrix}$

While two parameters are sufficient to parametrize a surface, objects of more than two dimensions will require more than two parameters. These objects, generally called manifolds, may live in higher than three dimensions and can have more than two parameters, so many times cannot be visualized. Nevertheless they can be analyzed using the methods of vector calculus and differential geometry.