Parametric Equations

Definition (Parametrized Curve)  Let be a curve defined by where and are defined on the interval The equations and for are parametric equations for with parameter

Definition (Orientation of a Curve)  The orientation of a parameterized curve is the direction determined by increasing values of the parameter.

Example (Orientation of a Parametrized Curve) Sketch the graph of the curve given by the following parametric equations by plotting values for and then check your graph by finding an equation in and only and then graphing. Show the orientation of the curve.

(a) on

    Solution. The graph starts at the point and follows the line until it reaches the other endpoint at
    



(b) on

    Solution. The graph starts at the point and follows the line until it reaches the other endpoint at
    


Comparing (a) and (b) we see that the graph of a curve may have more than one parametrization. Can you think of another set of parametric equations that gives the same graph?  

(c) on

    Solution. The curve starts at and follows the upper part of the unit circle until it reaches the other endpoint of
    
    

Can you think of another set of parametric equations that give the same graph?

Example (Parametric Equations for the Cycloid) The curve traced out by a fixed point on the circumference of a circle as the circle rolls along the top of the -axis is called a cycloid. Find parametric equations for a cycloid.

    Solution. Suppose the circle rolls along the -axis in the positive direction and has radius Here are some examples of the position that a circle will take as the circle rolls along on top of the -axis.
    
    


In order to obtain parametric equations consider the following diagram.



    Let represent the angle in radians. Thus, the distance the circle has rolled at time is where is the radius of the circle, by definition of arc length. So the coordinates at this point in time for is If we introduce the coordinate system and then and If represents an angle in standard position in the -plane, then Thus,





Now substitution into and yields





for any

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Cite this as:
Parametric Equations
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/parametric-equations.html