Equation of circle under different conditions

What is a circle?

A circle can be defined as the collection of points equidistant from another point. A circle can also be seen as an ellipse with its eccentricity being zero.

A circle is a 2 dimensional figure and we can represent it using an equation in the cartesian coordinate system and other orthogonal coordinate systems.

Circle’s equation in cartesian coordinate system:

The circle’s equation in the cartesian coordinate system depends on the coordinate of the centre and radius of the circle.

Let’s consider a circle of radius c with its centre at the point (0, 0). So the equation of such a circle will be given by, x²+ y²= c²

Now, let’s shift the centre of the circle from the origin to a random point (p, q). The equation will be, (x-p)² + (y-q)² = c²

Now let’s expand the above equation.

x² + y² + 2px + 2qy + (p² + q² – c²) = 0

Here the term in the bracket, (p2 + q2 – c2) is a constant, therefore, we can represent this using a constant, lets say k. Therefore the equation of the circle will now become,

x² + y² + 2px + 2qy + k = 0

This is a quadratic equation of two variables. Further this equation is a direction consequence of pythagoras theorem, sum of the square of the distance along both the axis gives us square of the radius.

Now let us take a look at a circle that is touching both x axis and y axis. Let the point where they touch the x axis be (p, 0) and that on y axis be (0, p). Then the centre of the circle will be (p, p).So therefore the equation of this circle will be given by,

x² + y² + (2x + 2y)p + c = 0

The radius of this circle will be r = 2p

The parametric form of circle’s equation.

The circle’s  equation in parametric form can be written as

x – p = c cosθ

y – q = c sinθ

The sum of there square will give us the equation of the form,

(x-p)² + (y-q)²= c²( cosθ² + sinθ²)

⇒ (x-p)² + (y-q)² = c²

Here θ is a parametric variable, the range of θ is from 0 to 2π.

We can also write this as x = p + c(1 – θ²)/(1 + θ²)

    y = q + c(2θ / 1 + θ²)

On simplifying this equation, we will get the same equation of the circle.

Equation of circle from the equation of an ellipse.

The equation of ellipse becomes the equation of a circle when the eccentricity becomes zero. i.e., e = 0. Therefore, e = ( 1 – (b/a)²)1/2. becomes ,

  e = ( 1 – (b/a)²)½ = 0

⇒    1 = (b/a)²

⇒    b = a ( a = semimajor axis, b = semiminor axis)

This gives both the axis to be equal.

We can now write the equation of this ellipse as 

(x/b)² + (y/b)²= 1

⇒   (x)² + (y)² = b²

Thus we can write the equation of a circle in a more generalised form using the equation of an ellipse. We can also say that a circle is an ellipse with both the semi-major and semi-minor axis being equal.

Equation of a circle in terms of polar coordinates.

In spherical polar coordinates the circle’s equation is given by,

r²– 2rdcos(θ – Φ) + d² = c²

Here c is the circle’s radius and r, θ, and Φ are the position coordinates in the spherical polar coordinate system. d is the distance between the centre of the circle and the origin. Here the circle’s centre is given by (r, Φ), and (r, θ ) gives us a point on the circle.

If the distance to the centre of  the circle, d becomes zero then the equation will be

   r² = c²

This means that the equation will get reduced to just the circle’s radius; the circle’s centre is in spherical polar coordinates. 

When the distance between origin and the circle’s centre becomes equal to the radius we get,

 r²– 2rdcos(θ – Φ) + d² = d²

⇒ r = 2dcos(θ – Φ)

Conclusion

A circle is a collection of points equidistant from another point. The circle is a 2-dimensional quantity. We can express the equation of a circle in different coordinate systems. The circle’s equation in the cartesian coordinate system is given by (x-p)2 + (y-q)2 = c2 ,where (p,q) is the circle’s centre. We can also express the circle as a special case of ellipse where the semi-major axis and semi-minor axis are both equal. Sometimes curvilinear coordinates such as spherical polar coordinates are used to express the equation of motion. Then the equation of motion becomes, r2– 2rdcos(θ – Φ) + d2 = c2. And when the centre of the circle coincides with the origin we get the equation of the circle to be as simple as r = c.