Dated: 30-10-2024

Polar Coordinate Systems

To form a polar coordinate system, we take a fixed point \(O\) called origin or pole.
Then we construct a ray in right direction from it called a polar axis.
After selecting a unit for measurement, we may associate any point \(P\) in the plane¹ with the pair \((r, \theta)\).
Here \(r\) is the radial distance from \(O\) to \(P\) and \(\theta\) is the angle between polar axis and the line² \(\overline{OP}\) called polar angle.

Uniqueness

The polar coordinates of a point are not the same.
A point may be represented as following

\[(5, 45 \degree)\]

\[(5, 405 \degree)\]

\[(5, -335 \degree)\]

They all represent the same point.

\[(r, \theta + 360 \degree \cdot n) \text{ where } n \in \mathbb{Z}\]

Negative Values for \(r\)

Although distance being negative does not make sense but we can have negative values for \(r\).
Here is an example

It just means that if we look at \(r\) as a vector³ \(\vec{r}\), we are just flipping the signs of its components.

Relation between Polar and Cartesian Coordinate System

\[x = r \cdot \cos(\theta)\]

\[y = r \cdot \sin(\theta)\]

\[\tan (\theta) = \frac y x\]

`Lines`⁴ In Polar Coordinate System

Horizontal line

\[\because y = a \text{ and } y = r \cdot \sin(\theta)\]

\[r \cdot \sin(\theta) = a\]

Vertical line

\[\because x = b \text{ and } x = r \cdot \cos(\theta)\]

\[r \cdot \cos(\theta) = b\]

Along \(\overline{OP}\)

\[\theta = \theta_{\circ}\]

General Equation

The general equation for the line⁴

\[Ax + By + C = 0\]

will take the form

\[A(r \cdot \cos(\theta)) + B(r \cdot \sin(\theta)) + C = 0\]

\[r \left(A \cdot \cos(\theta) + B \cdot \sin(\theta)\right) + C = 0\]

Circle in Polar Coordinates

We know that the equation for a circle centered at \((x_0, y_0)\) with radius \(a\) is

\[a^2 = (x - x_0)^2 + (y - y_0)^2\]

Converting \(x\) and \(y\) into their polar forms, we get

\[a^2 = (r \cdot \cos(\theta) - r_{\circ} \cdot \cos(\theta_{\circ}))^2 + (r \cdot \sin(\theta) - r_{\circ} \cdot \sin(\theta_{\circ}))^2\]

Expand this and we will get

\[r^2 - 2 r \cdot r_{\circ} \cdot \cos(\theta - \theta_{\circ}) + r_{\circ}^2 = a^2\]

Special Cases for `Circle`

If the circle is centered at the origin then the equations becomes

\[r^2 = a^2 \implies r = a\]

However, the circles with polar coordinates \((a, \frac \pi 2)\) and \((a, - \frac \pi 2)\) are

In these cases, the equations become

\[r = 2 a \sin(\theta)\]

\[r = -2 a \sin(\theta)\]