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, as shown in this page's main image.

This curve uses the following parametrization:

Parametrized Curves

Many useful or interesting shapes otherwise inexpressible as xy-functions can be represented in coordinate space using a non-coordinate parameter, such as circles. A circle cannot be expressed a function where one variable is dependent on another. If a parameter (t) is used like to represent an angle in the coordinate plane, the parameter can be used to generate a unit circle, as shown below. The parameter does, in the case of a unit circle, represent a physical quantity in space: the angle between the x-axis and a vector of magnitude 1 going to point on the coordinate plane.

Parametrized Surfaces

The surface of a sphere can be graphed using two parameters.

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:

Parametrized Manifolds

While two parameters are sufficient to parametrize a surface, objects of more than two dimensions, such as a three dimensional solid, 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.

Parametric Equation Explorer

This applet is intended to help with understanding how changing an alpha value changes the plot of a parametric equation. See the in-applet help for instructions.

If you can see this message, you do not have the Java software required to view the applet.