Polar Coordinate System

1. Polar Coordinate System is a 2 Dimensional Curvilinear Coordinate System.
2. The Radial Axis and the Tangential Axis form the 2 Coordinate Axes of the Polar Coordinate System.
   
   The Straight Lines Emerging Radially from a Reference Point form the Straight Lines along the Radial Axis. The Concentric Circles having Center as the same Reference Point form the Curves along the Tangential Axis.
   
   This Reference Point is called the Pole and forms the Origin / Center of the Polar Coordinate System.
3. The Grid for the Polar Coordinate System is formed by the Intersections of the Straight Lines along the Radial Axis and the Curves (Concentric Circles) along the Tangential Axis.
   
   
   
   Each such Point of Intersection specifies a Distinct Location on the Grid of the Plane represented by Polar Coordinate System known as a Coordinate Location or a Coordinate Point and is given by Unique Ordered Pair of Real Number Values (\(r, \theta\)).
   
   The First Value of the Ordered Pair, (i.e. \(r\)), gives the value of the Radial Axis which depicts the Distance of the Coordinate Point from the Pole / Origin / Center.
   
   The Second Value of the Ordered Pair, (i.e. \(\theta\)), gives the value of the Tangential Axis which depicts Counter Clockwise Angle that the Coordinate Point has from a Reference Direction. This Reference Direction is typically same as the Positive Direction of \(X\) Axis of the 2 Dimensional Cartesian Coordinate System.
4. At any given Point of Intersection, the Straight Lines along the Radial Axis are Perpendicular to the Tangents to the Curves (Concentric Circles) along the Tangential Axis. Hence, the Polar Coordinate System is also an Orthogonal Coordinate System.
5. Any Coordinate Point (\(r, \theta\)) given in Polar Coordinate System can be converted to a Coordinate Point (\(x,y\)) in 2 Dimensional Cartesian Coordinate System as follows
   
   \(x=r \cos\theta\hspace{7mm}y=r \sin\theta\)
   
   Conversely any Coordinate Point (\(x,y\)) given in 2 Dimensional Cartesian Coordinate System can be converted to a Coordinate Point (\(r, \theta\)) in Polar Coordinate System as follows
   
   \(r=\sqrt{x^2 + y^2 }\hspace{7mm}\theta=m\_arctan2\hspace{1mm}(x,y)\)
   
   The \(m\_arctan2\) is the Modified Arctangent Function , which calculates the angle \(\theta\) such that \(0 \leq \theta < 2\pi\).