Plane Curves

Definition (Plane Curve)  A plane curve is a set of ordered pairs of functions and defined on an interval Often the term curve is used instead of plane curve.

Definition (Endpoints of a Curve)  Let be a curve with then and are called the endpoints of

Definition (Closed Curve)  A plane curve such that where is defined on is called a closed curve.

Definition (Simple Closed Curve)  A plane curve that does not intersect itself and such that where is defined on is called a closed curve.

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

Example (Parametrized Curve) Find an equation in and whose graph contains the points on the curve

(a) for
    
    Solution. Recalling we square and add: Here is a graph of the curve:



    
(b) for
    
    Solution. Since the exponents sum to 0, we have and so with Here is a graph of the curve:


    

(c) for
    
    Solution. We have Here is a graph of the curve:


    

(d) for

    Solution. We have Here is a graph of the curve:


    

(e)   for and

    Solution. We have and so we have a hyperbola centered at
    Here is an example with We have and so the hyperbola is centered at the origin and has vertices The asymptotes are   



    Using the graphs of the secant and tangent functions we can notice the following:
    
    If then and ; so we have the lower right branch of the hyperbola.
    If then and ; so we have the upper right branch of the hyperbola.
    If then and ; so we have the lower left branch of the hyperbola.
    If then and ; so we have the upper left branch of the hyperbola.     

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