Parametric Equations
Parametric equations are those that relate typical x and y values to another variable or arbitrarily chosen constant. Such equations are widely found in studies dealing with motion as a function of time. There are scalar and vector parametric equations. In vector parametric equations the arbitrary variable is called a parameter and it is related to a vector via an ordered pair. Scalar parametric equations are more commonly referred to as just parametric equations. They are the set of equations that relates the arbitrarily chosen constant or variable to the traditional Cartesian coordinates x and y. For example let us consider a cycloid that is formed as a point on a circle traces out a curve as the circle rolls along a line. Although one can represent the cycloid as the graph of a function it is also possible to represent the curve C using parametric equations. In this case the point (x,y) on C can be expressed as functions of time t: x = f(t) and y = g(t), where f(t) and g(t) are continuous functions on an interval such that C consists of all points (x,y). In this case the equations noted above are called parametric equations of C and we say that C is parametrized by those equations with t being a parameter of C.
The vector parametric equation r = tv + b can be interpreted as giving the position r at time t of a particle that is at b when t = 0 and moves with a constant vector velocity v.
This interpretation yields a graph of the equation that is called the trajectory of the particle. These equations are only applicable to vector situations.
The more commonly used type of parametric equation is the scalar parametric equation, which from here onwards will be referred to simply as parametric equations. Relating parametric equations to the corresponding Cartesian equation is often times a simple task. Take for example the parametric equations x = sinΘ, y = cos2Θ. They can easily be converted to the corresponding Cartesian form: y = cos2Θ = 1 - 2sin2Θ = 1 - 2x2. Although the Cartesian equation can be obtained from parametric equations by eliminating the parameter from the set of parametric equations it is often not equivalent to the original set of parametric equations. This is because the coordinates of points that are not given by the parametric equations may satisfy the Cartesian equation. In order to obtain equivalent parametric and Cartesian equations any restrictions on the domain that may be implicit in the parametric form of the equation but do not appear in the Cartesian form must also be stated explicitly. In the previous example it is clear that for x = sinΘ that -1 x 1. As Θ takes on all real values in the parametric equations it is clear that the curve traces and retraces the arc of y = 1 - 2x2 that is bounded by (-1, -1) and (1, -1). The Cartesian equation should therefore be stated with the restriction -1 x 1.
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