Parametric Equations can be defined as equations which are expressed using some parameters or we can say that the parametric equation is used to define a relation using parameters. For Example: Equation of circle in Cartesian coordinates is defined as x^{2} + y^{2} = r^{2} and parametric equation of circle will be:
x = r cos t,
y = r sin t,
Here 'x' and 'y' are parameters of circle.
We can say that parameters of a parametric equation are just like parameter time, which is used to find out position, velocity etc. These equations give value of parameters of an equation.
Let us see some examples of parametric equations:
Parabola: We know that simplest way to represent a parabola is y = x^{2}. This is a Cartesian equation of parabola then parameter of parabola will be 'x' and 'y'. We can find parameters of parabola by parametric equations:
x = t,
y = t^{2},
Here equation is represented in terms of parameter 't'. Parametric equation can also be used to express 3-D objects.
Let us take an example of helix. Helix needs three parameters to define its position in space. We can find out parameters of helix using parametric equation. We need to find out three parameters x, y and z, equations are:
x = a cos t,
y = a sin t,
z = bt,
Basically parametric representation is not unique in nature that means we can represent these equations in many ways. Quantity can be represented by several parameterizations.
Mathematical figures (curves) are usually explained by graphing parametric equations. For instance, suppose we have a geometrical figure which has been well-defined with the help of its equation in rectangular coordinate system or 2 – D plane. x and y- coordinates can be represented as functions of some parameter “c”. In order to get original equ...Read More