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.
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 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.
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.
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)
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.
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