Polar Cylindrical Coordinate System

1. Polar Cylindrical Coordinate System is a 3 Dimensional Curvilinear Coordinate System.
2. Polar Cylindrical Coordinate System is formed by Extending the Polar Coordinate System to 3 Dimensions by Adding a Coordinate Axis (called the \(Z\) Axis) Perpendicular to the Plane of the Polar Coordinate System passing throught its Origin / Center.
   
   Thus, the Polar Coordinate System forms the Base of the Polar Cylindrical Coordinate System. The Origin / Center of the Polar Coordinate System also serves as the Origin of the Polar Cylindrical Coordinate System.
   
   The Radial Axis and Tangential Axis corresponding to the Polar Coordinate System along with the \(Z\) Axis form the 3 Coordinate Axes of the Polar Cylindrical Coordinate System.
3. The Grid for the Polar Cylindrical Coordinate System is formed by the Intersections of the Straight Lines along the Radial Axis, the Curves (Concentric Circles) along the Tangential Axis and the Straight Lines Parallel to the \(Z\) Axis.
   
   Each such Point of Triple Intersection specifies a Distinct Location on the Grid of the Space represented by Polar Cylindrical Coordinate System known as a Coordinate Location or a Coordinate Point and is given by Unique Ordered Set of 3 Real Number Values (\(\rho, \theta, z\)).
   
   The First Value of the Ordered Set (i.e. \(\rho\)), gives the value of the Radial Axis which depicts the Perpendicular Distance of the Coordinate Point from the \(Z\) Axis.
   
   The Second Value of the Ordered Set, (i.e. \(\theta\)), gives the value of the Tangential Axis which depicts the 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.
   
   The Third Value of the Ordered Set, (i.e. \(z\)), gives the value of the \(Z\) Axis which depicts the Signed Perpendicular Distance of the Coordinate Point from the Base of the Polar Cylindrical Coordinate System.
4. At any given Point of Intersection, the Straight Lines along the Radial Axis, the Tangents to the Curves (Concentric Circles) along the Tangential Axis and the Straight Lines Parallel to the \(Z\) Axis are Mutually Perpenticular to Each Other. Hence, the Polar Cylindrical Coordinate System is also an Orthogonal Coordinate System.
5. Any Coordinate Point (\(\rho, \theta,z\)) given in Polar Cylindrical Coordinate System can be converted to a Coordinate Point (\(x,y,z\)) in 3 Dimensional Cartesian Coordinate System as follows
   
   \(x=\rho \cos\theta\hspace{7mm}y=\rho \sin\theta\hspace{7mm}z=z\)
   
   Conversely any Coordinate Point (\(x,y,z\)) given in 3 Dimensional Cartesian Coordinate System can be converted to a Coordinate Point (\(\rho, \theta,z\)) in Polar Cylindrical Coordinate System as follows
   
   \(\rho=\sqrt{x^2 + y^2 }\hspace{7mm}\theta=m\_arctan2\hspace{1mm}(x,y)\hspace{7mm}z=z\)
   
   The \(m\_arctan2\) function that calculates the value of \(\theta\) is the Modified Arctangent Function.