Parametric equations of a circle

A circle can be a set of all curvy points equidistant from a specific point called the circle’s centre. This distance between the circle and its centre is known as radius. Parametric equations of a circle help define the parameters and coordinating points of a circle. The parameters of a circle are independent variables. Furthermore, the circle’s parametric equations help us represent the curves on the graph, respectively. 

Parametric Equations of a Circle

The parametric equations help to describe curves in the two-dimensional coordinate system. Most of the time, these parameters are not functions but are independent variables.

Parametric Equations for a Standard Circle

Let us consider the circle’s centre as point O, and line OP is the radius equal to r. It has its centre at the origin (0, 0). In Cartesian coordinates, the equation of a circle with a point (x, y) on it is represented as follows:

x2 + y2 = r2

Let us consider the point P (x, y) as the coordinates of any point on the circle. We get a right-angled triangle when a perpendicular line is drawn from the point P to the X-axis. Here, an acute angle is formed opposite to the perpendicular line represented as θ. It is called a parameter.

With the help of trigonometric ratios, we get the following values of x and y:

y = r sin θ and x = r cos θ.

Therefore, the parametric equation of a circle that is centred at the origin (0,0) can be given as P (x, y) = P (r cos θ, r sin θ),

(Here 0 ≤ θ ≤ 2π.)

In other words, it can be said that for a circle centred at the origin, x2 + y2 = r2 is the equation with y = r sin θ and x = r cos θ as its solution. Here, θ is the parameter.

For example, if a circle centred at the origin has its radius of 10 cm, the solution of its parametric equation will be x = 10 cos θ and y= 10 sin θ.

Parametric Equations for a General Circle

For a general circle, that is, a circle not centred at the origin. We just need to add or subtract a certain fixed value to the coordinates x and y.

Let us consider (h, k) as the coordinates of the centre of the circle. Here, we just need to add them to the coordinates x, and y coordinates in the equation, same as that for the circle centred at the origin. Hence, to get the parametric equation, we need to add the coordinates h and k to x = r cos θ, y = r sin θ.

Therefore, the parametric equation of the circle that is not centred at the origin of the circle but the coordinates (h, k) can be written as follows:

x = h + r cos θ and y = k + r sin θ,

(Here, 0 ≤ θ ≤ 2π.)

Applications of Parametric Equations

The parametric equations have tremendous applications in a variety of fields. Some of them are as follows:

• Kinematics

The parametric equations play a key role in kinematics. Here, the paths of the objects through the space can be commonly described as parametric curves. In kinematics, the parametric equations for the coordinates of the given object collectively form a vector-valued function. These curves are further differentiated and integrated termwise.

• Computer-aided Design

Parametric equations are also used in the field of computer-aided design. For instance, parametric equations allow the users to enjoy the benefit of both the implicit and explicit representations. They help in generating and plotting points on a curve easily and effectively.

• Integer Geometry

In integer geometry, various problems can be solved by referring to parametric equations. One of the widely used examples is Euclid’s parameterization for the right-angled triangles.

Conclusion

The term ‘circle’ can be defined as a collection of curvy points with a radius equidistant from the centre to the curve of the circle. The parametric equations are one of the most important topics in geometry. Using the parametric equations, we can define the points and coordinates on curves.  In short, we can define the coordinates on a circle, hyperbola, parabola, and ellipse. The parametric equations for a standard circle and general circle are different as the standard circle is centred at the origin while the general circle is not centred at the origin. Hope this parametric equations study material helps crack your exams with ease.