Parametric Equations

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Butterfly Curve
The Butterfly Curve is one of many beautiful images generated using parametric equations.


Basic Description

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 consider functions where each coordinate is equal to an equation of an independent variable, known as a parameter. Changing the value of the parameter can change the 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. Parametrization is the process of finding a parametrized version of a function.

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 it 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:

The butterfly curve in this page's main image uses a similar method, but with more complicated parametric equations.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Linear Algebra

Sometimes curves which would be very difficult or even impossible to graph in terms of elementary fun [...]

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, as shown in this page's main image.

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}

Parametric construction of the butterfly curve
Parametric construction of the butterfly curve

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 cannot always be visualized. Nevertheless they can be analyzed using the methods of vector calculus and differential geometry.

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