Parametric Equation of Circle

In our other article, we have learned about the standard form and the general form of the equation of the circle. Here, in this article, we will deal with the other form, the parametric form of the equation of a circle.

Parametric Equation of Circle – Center at Origin (0, 0)

We know, the equation of a circle in Cartesian coordinates, centered at the origin (0, 0) and having a point (x, y) on the circle is given by x² + y² = r²

Similar to the parametric equation of a line, the parametric equation of a circle will help us to find the coordinates of any point on a circle centered at the origin (0, 0) with radius ‘r’. See our Equation of a Circle article for reference.

Let P (x, y) be the coordinates of any point on the circle. If we draw a perpendicular line from point P to the x-axis meeting at S, we get a right triangle.

Thus, triangle OPS is a right triangle, where OS is the base of the triangle, SP is the perpendicular of the triangle, and θ is the angle made by OP with the x-axis is called a parameter.

From the basics of trigonometry,

OS/OP = cos θ

=> OS = OP cos θ …… (1)

Similarly,

SP/OP = sin θ

=> SP = OP sin θ …… (2)

Since, OS = x, SP = y, OP = r

Substituting these values in equation (1) and (2), we get

x = rcos θ, y = rsin θ

Thus, the parametric equation of the circle centered at the origin is written as P (x, y) = P (r cos θ, r sin θ), where 0 ≤ θ ≤ 2π. See Fig.1 (a) in the below-given diagram.

In other words, for all values of θ, the point (rcosθ, rsinθ) lies on the circle x² + y² = r². Or, any point on the circle is (rcosθ, rsinθ), where θ is a parameter.

Let us take an example to understand the concept better.

If we have a circle of radius 10 units and with its center at the origin, the circle can be described by the pair of equations

x = 10 cos θ, 10 sin θ

What Does Parametric Mean

As discussed above, the angle θ (theta) is called a parameter, which represents the angle made by the line joining the point (x, y) with the center, and also with the x-axis.  This is just a variable that appears in a system of equations that can have any value (unless specified) but has the same value everywhere it is used. Thus a circle equation represented in the form x = r cos θ, r sin θ is called the parametric equation of a circle.

Now, let us now derive the parametric equation of a circle not centered at the origin.

Parametric Equation of Circle – Center at (h,k)

This is simple. We just need to add or subtract fixed amounts to the x and y coordinates. If (h, k) be the coordinate of the center of the circle, we simply add them to the x and y coordinates in the above equation x = r cos θ, y = r sin θ, to get the equation.

Thus, the parametric equation of the circle centered at (h, k) is written as,

x = h + r cos θ,  y = k + r sin θ, where 0 ≤ θ ≤ 2π