Curves Defined by Parametric Equations



In this section we will study situations in which three variables are used to represent a curve in the plane.


Suppose that x and y are both given as continuous functions of a third variable t (called a parameter) by the equations



(called parametric equations). Each value of t determines a point , which we can plot in the coordinate plane. As t varies, the point  varies and traces out a curve C, which we call a plane curve or parametric curve.


Parametric graphing is like watching the curve draw itself over time. It’s spiritual, man.



As an example, consider the path followed by an object that is propelled into the air at an angle of . If the initial velocity of the object is 48 feet per second, the object travels the parabolic path given by


By writing x and y as functions of time t, we can obtain parametric equations



When will the object reach maximum height?



As another example, sketch and identify the curve defined by the parametric equations



What if we restrict so that ?



As further examples, what curves are represented by the parametric equations below?




Use a graphing device to graph the curve . (Note that in general, we can graph curves of the form  parametrically by letting  and .)



Eliminating the Parameter



Finding a rectangular equation that represents the graph of a set of parametric equations is called eliminating the parameter. The range of x and y implied by the parametric equations may be altered by the change to rectangular form.


As an example, sketch the curve represented by the equations below by eliminating the parameter and adjusting the domain of the resulting rectangular equation.




Finding Parametric Equations



Find a set of parametric equations to represent the graph of , using each of the following parameters.



As you can see, a parametric expression need not be unique.






The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid. Find parametric equations for the position for the cycloid.



Smooth Curves



A curve C represented by  and  on an interval I is called smooth if  and  are continuous on I and not simultaneously 0, except possibly at the endpoints of I.