1.1.1 Introduction to parameterized curves
A parameterized curve is a function from a subset of into Euclidean space (). For the time being we are going to stick to plane curves, meaning our codomain is .
Notation:
where .
The input variable is often called the parameter. The functions and are called the component functions of . The function itself is called a path in and the image of the function is called the image curve of the path. The points and are the endpoints of the curve.
The first thing I want you to do is just play. In the GeoGebra applet below you can type in two component functions and a domain of definition to see the resulting parameterized curve. Take some time to experiment. Try fixing a couple component functions and changing the domain to see how the resulting curve changes. Then hold the domain steady and edit the component functions creating different curves. Can you parameterize a line segment? A circle? A spiral? Can you create a curve whose endpoints are the same? How about a curve with a loop-de-loop?