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Unformatted text preview: MATH 118, LECTURES 11: PARAMETRIC EQUATIONS 1 Parametric Equations We are used to seeing functions written in the form of, for example, y = f ( x ) which expresses that y depends explicitly on x . When each x determines a unique value of y (verticle line test), we say that y is a function of x ; otherwise, we say that x and y are relational and the equation must be given implicitly (as in, say, x 2 + y 2 = 1). That is to say: Representation Relationship General Form Explicit Function y = f ( x ) Implicit Relation f ( x, y ) = 0 For many applications, these representations are not ideal. For example, imagine being asked to get a computer to print a box with vertices at (1 , 1), (- 1 , 1), (- 1 ,- 1), and (1 ,- 1) (see Figure 1). We could not represent our input in the form y = f ( x ); in fact, even if we could input the four sides separately, the equation y = f ( x ) still does not tell the computer we are interested in only the portion of the line bound by the vertices, and not the infinite line passing through it. Similarly, functions (or relations) with irreg- ularities (such as cusps, intersections, etc.) are not generally well handled by either the form y = f ( x ) or the implicit for f ( x, y ) = 0. We need another approach, and fortunately we have one. Imagine, in representing our box, we simply assigned points on each side to an arbitrary parameter , say, t . Lets suppose, for simplicity, that we are connecting the vertex (- 1 , 1) to (1 , 1) and we wish to associate t = 0 to the point (- 1 , 1) and t = 1 to (1 , 1). How can we do this? The only coordinate that changes is the first one, so we can readily see that (- 1 + 2 t, 1) , t 1 satisfies both of the endpoint requirements, and connects the endpoints with a straight line. Furthermore, we have clearly indicated through the bounds on the parameter that we want the line to stop at the vertices! But what have we really done here? We could have written this as x ( t ) =- 1 + 2 t, y ( t ) = 1 , t 1 . 1 So, rather than having x and y depend on each other, they both depend on an independent parameter, which we have called t . This is called parametriza- tion , and the equations which result (like the one above) are called paramet- ric equations . We can complete the other three sides to get the equations x ( t ) =- 1 + 2 t, y ( t ) =- 1 , t 1 x ( t ) =- 1 , y ( t ) =- 1 + 2 t, t 1 x ( t ) = 1 , y ( t ) =- 1 + 2 t, t 1 . Together with the previous equation, we have a complete description of the box in Figure 1 (in fact, these are the exact equations I used to code it). Figure 1: A box with vertices (1 , 1), (- 1 , 1), (- 1 ,- 1), and (1 ,- 1). So the basic idea with parametrization is that, instead of having x and y relate to one another directly, we set up an independent variable (called the parameter ) which both x and y depend on. So, instead of y = f ( x ) or f ( x, y ) = 0 we have y = y ( t ) , x = x ( t ) , t I where I is some interval (possibly the whole real number line).(possibly the whole real number line)....
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- Spring '09