Parametric Equation

All the curves cannot be represented by the equation, where y is a function of x. In such cases, we need an approach that allows us to represent both the x and y coordinates in terms of a third variable. This new variable introduced is known as a parameter. More formally, we can say that the Parametric equation belongs to a class of equations that employs an independent variable called a parameter. Here the dependent variables are defined as continuous functions of the parameter and they are not dependent on any other existing variable. Instead of that, given a pair of parametric equations with parameter t, the set of points (f(t), g(t)) forms a curve in the plane. In this article, we will study different types of parametric equations of the curve like the parametric equation of circle, line, plane, ellipse, parabola and hyperbola.

More than one parameter can be employed whenever required. This process of conversion to parametric form is called parameterization. For some complicated curves, parametrization provides great efficiency while differentiating and integrating the curves. Generally, a single parameter is represented with the parameter t,  and if there are 2 parameters involved, then the commonly used symbols are u  and v. 

Example of a Parametric Equation:

The equations x = t and y = t² are parametric equations for the curve y=x².

Note that parametric representations for a curve are generally not unique.  The same quantities can be expressed by a number of different parameterizations. In the above example, the equations x = (1+u) and y = (1+u)²  can also be an example of parametric equations.

Parametric Equation of Circle

Parametric Equation for the Standard Circle:

Consider the equation of the circle, whose centre is at O(0, 0) and radius is r.

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Let’s consider the point  P(x, y) on the circle such that the OP makes an angle θ with the X-axis. Using trigonometry, we’ll get:

x = rcosθ

y = rsinθ

This is the parametric form of circle. Here, θ is the parameter that represents the angle made by line OP with the X-axis.

In other words, for all values of θ, the point (rcosθ, rsinθ) lies on the circle \[x^{2}+y^{2}=r^{2}\] or, any point on the circle is (rcosθ, rsinθ), where θ is a parameter.

We can also find the parametric equation of the circle whose centre is not located at the origin.

Parametric Equation for the General Circle:

Consider the General Equation of the Circle:

\[x^{2}+y^{2}+2gx+2fy+c = 0\]

This can be written as:

\[(x+g)^{2}+(y+f)^{2} = r^{2}\] (where r² = g² + f²- c)

Again, consider point P(x, y) to be any point on the circle such that CP makes an angle θ with the X-axis.

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In this case, we’ll get:

x + g = rcosθ

y + f = rsinθ

This gives us

x = –g + rcosθ

y = -f + rsinθ

And, this is the parametric equation of the circle \[x^{2}+y^{2}+2gx+2fy+c = 0\] 

Parametric Function

Parametric functions represent the number of coordinates (for example 2 for the 2-dimensional plane, 3 for 3-D space, and so on), where each of coordinate (x, y, z ...) is expressed as another function of some parameter, like time: x = f(t), y = g(t), z = h(t), and so on.

Suppose function is y=f(x),  

Parametric form will be x=g(t) , y=h(t)

Parametric Curve

The curves which are described by parametric equations are known as parametric curves. With the help of parametric equations, we can describe all types of curves that can be represented on a plane. They are mostly used in situations where curves on a Cartesian plane cannot be described by given functions (e.g., when a curve crosses itself). It is also often used in three-dimensional spaces, with more than three dimensions by implementing more parameters.

Parameterizing Curves

If the given Cartesian equation expresses one variable as a function of the other, then parameterization is quite easy. In such cases, the independent variable in the function can simply be defined as the function of t. 

Considering the cartesian equation y = x² - 3x.

Here, x is the independent variable. So taking x = t and substituting it in the cartesian equation, we get,

y = t² - 3t. Combining these equations, we have x =t and y = t² - 3t as the parametric equation. Thus we have parameterized the curve.

Converting from Parametric to Cartesian

In some cases, it is possible to eliminate the parameter t from the parametric equations, thus allowing you to write parametric equations as a Cartesian equation. It is easiest to do this conversion if one of the x(t) or y(t) functions can easily be solved for t, as we can substitute the value of the parameter t in the other equation to find the cartesian equation. 

Consider the parametric equation x(t) = t² + 1 and y(t) = 4 + t.

Here, y(t) is the linear function of t, so we can easily solve for t. We have t = y - 4. Now, substituting the value of t in x(t) we get x =  (y - 4)² + 1, which is the cartesian equation for the given parametric equations.

To check the Cartesian equation to be as equivalent as possible to the original parametric equation, we try to avoid using domain-restricted inverse functions, such as the inverse trigonometric functions. For equations involving trigonometric functions, we often try to find an identity to avoid the inverse functions.

Parametric Equation of a Line

The parametric equation of a straight line passing through (\[x_{1},y_{1}\]), making an angle θ with the positive X-axis is given by \[\frac{x-x_{1}}{cos\theta} = \frac{y-y_{1}}{sin\theta} = r \], where r is a parameter, which denotes the distance between (x, y) and  (\[x_{1},y_{1}\]).

Parametric Equation of Ellipse

Standard Equation of Hyperbola: \[\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\] 

As we know, coordinate point x = a cost and y = bsint satisfy the hyperbola equation for all real values of t. Therefore x = acost, y = bsint are the parametric equation of hyperbola \[\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\] .

Where the parameter t lies between \[ 0 \leq t \leq 2 \pi \] .

Parametric Equation of Parabola

Here we will discuss the parametric equation of parabola\[y^2 = 4ax \]. The parabola \[y^2 = 4ax \] is a lot of times specified not in the standard x – y form but instead in a parametric form, in terms of a parameter t.

The equation \[y^2 = 4ax \] can be equivalently written in parametric form \[x = at^{2}, y = 2at \].

This is easily verifiable by substitution. Thus, for any value of t, the point \[(at^{2},2at)\] will always lie on the parabola \[y^2 = 4ax \]. Different values referred to as simply the point give rise to different points on the parabola.

Parametric Equation of a Plane

A plane in three dimensions is determined by a point (a, b, c) on the line and a direction \[\bar v \]that is parallel to the line.

Set of points on this line is given by \[ (x,y,z)=(a,b,c)+ (t \bar{v} , t \in R )\]

The equation (x, y ,z) = (a, b, c) + t\[\bar v\] is called the vector equation of the line.

We can also rewrite this as three separate equation: if vector\[v = (v_{1},v_{2},v_{3})\] then (x,y,z) is on the line, if 

\[x = a+tv_{1}\]

\[y = a+tv_{2}\]

\[z = a+tv_{2}\]

are satisfied by the same parameter \[t \in R\] . This is called the parametric equation of the line.

Parametric Equation of Hyperbola

Standard Equation of Hyperbola: \[\frac{x^{2}}{a^2} - \frac{y^2}{b^2} = 1\]

As we know, coordinate points \[x= asec \theta\] and \[y = btan\theta\] satisfy the hyperbola equation for all real values of \[\theta\]. Therefore, \[x= asec \theta\],\[y = btan\theta\] are the parametric equation of hyperbola \[\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\],

Where the parameter \[ 0 \leq \theta \leq 2\pi \].

The angle \[\theta\] is called eccentric angles of point \[(x= asec\theta,y = btan \theta)\] on the hyperbola.