Intro to Plane Curves & Parametric Equations - Calculus 2

In the last lesson we completed the second major group of concepts in Calculus 2, which was sequences and series. With the conclusion of those concepts, we are now ready to begin the third and final major group of concepts in Calc 2, which is parametric equations and polar coordinates. 

We will begin with parametric equations, but before we discuss what parametric equations are, we need to establish why we even need them in the first place. Parametric equations are used to represent what are known as plane curves.

Simply put, plane curves are the resulting path from the motion of an object in a plane (or in two dimensions). Naturally, since a plane curve is the path of an object, there must be a starting point and ending point for this object, as well as a direction that describes where the object is heading. Because of these specific characteristics of plane curves, it doesn't make sense to represent them with regular functions in terms of x and y, or what we call rectangular equations. This is the case for a couple of reasons:

1. The resulting path from the motion of the object has the potential to create a shape that cannot be defined as a function (the plane curve would fail the vertical line test).
2. Rectangular equations cannot describe where an object begins or ends, or where it may be at any particular point in time.
3. Rectangular equations do not provide a direction. 

Because of these reasons, we can't use equations in terms of just x and y to accurately represent plane curves. Instead what we need to do, is represent x and y in terms of another variable, which we will call the parameter. This creates what are known as parametric equations.

Notice that the word "equations" is plural. This is important because parametric equations often come as a pair or a set: one equation for x in terms of the parameter, and another equation for y in terms of the parameter. Essentially what is happening, is the motion of an object is being defined in the x and y directions as a function of the parameter.

We typically use t as the parameter for parametric equations, but really any letter could be used. You will see some examples that use θ, particularly with parametric equations involving trig functions, but the most common letter used as the parameter will be t. We typically view the parameter as representing "time," but this doesn't mean that t can't be negative. Therefore it may be easier to view the parameter as just representing a sequence of points that make up the path of motion for an object.

Parametric equations fix all the issues that rectangular equations had with representing plane curves. A set of parametric equations can indicate a direction, as well as the starting point (called the initial point) and the ending point (called the terminal point) for a plane curve, as the equations are defined for a common domain of values for t. Therefore, we will be able to use a set of parametric equations to accurately sketch the graph of a plane curve. This leads us to the main goals of this lesson which are listed below. 

In this lesson, you will learn:

• What a plane curve is and why they need to be represented with parametric equations
• How to sketch a graph of a plane curve by creating a table and/or eliminating the parameter
• How to find a set of parametric equations for a given rectangular equation

I hope you find this video to be helpful!

-Josh