Spherical Coordinate System

1. Spherical Coordinate System is a 3 Dimensional Curvilinear Coordinate System.
2. The Radial Axis, the Polar Axis and the Equatorial Axis form the 3 Coordinate Axes of the Spherical Coordinate System.
   
   The Straight Lines Emerging Radially from a Reference Point assumed to be Center of a Sphere form the Straight Lines along the Radial Axis. The Semi-Circular Curves having Center as the same Reference Point emerging from Northernmost Point of Sphere (North Pole) to the Southernmost Point of Sphere (South Pole) form the Curves along the Polar Axis. The Circlular Curves Perpendicular to Semi-Circular Curves along Polar Axis Parallel to Equator of the Sphere form the Curves along the Equatorial Axis.
   
   The Reference Point (Center of the Sphere) forms the Origin / Center of the Spherical Coordinate System.
3. The Grid for the Spherical Coordinate System is formed by the Intersections of the Straight Lines along the Radial Axis, the Curves (Semi-Circles) along the Polar Axis and the Curves (Circles) along the Equatorial Axis.
   
   Each such Point of Triple Intersection specifies a Distinct Location on the Grid of the 3 Dimensional Space represented by Spherical Coordinate System known as a Coordinate Location or a Coordinate Point and is given by Unique Ordered Set of 3 Real Number Values (\(r,\theta,\phi \)).
   
   The First Value of the Ordered Set, (i.e. \(r\)), gives the value of the Radial Axis which depicts the Distance of the Coordinate Point from the Origin / Center.
   
   The Second Value of the Ordered Set, (i.e. \(\theta\)), gives the value of the Polar Angle which depicts Angle that the Coordinate Point has from the Direction of North Pole of the Sphere (Also known as Polar Axis).
   
   The Third Value of the Ordered Set, (i.e. \(\phi\)), gives the value of the Equatorial Angle which depicts Counter-Clockwize Angle that the Coordinate Point has from a Reference Direction on the Equatorial Plane (i.e a Plane Perpendicular to the Polar Axis of the Sphere).
4. At any given Point of Intersection, the Straight Lines along the Radial Axis, the Tangents to the Curves (Semi-Circles) along the Polar Axis and Tangents to the Curves (Circles) along the Equatorial Axis are Mutually Perpendicular to each other . Hence, the Spherical Coordinate System is also an Orthogonal Coordinate System.
5. Any Coordinate Point (\(r, \theta, \phi\)) given in Spherical Coordinate System can be converted to a Coordinate Point (\(x,y,z\)) in 3 Dimensional Cartesian Coordinate System in the following ways
   1. When the Polar Angle \(\theta\) is measured from the North Pole of the Sphere that lies on Positive Direction of \(Z\) Axis (which serves as the Polar Axis) and the Equatorial Angle \(\phi\) is measured from the Positive Direction of \(X\) Axis (Such that the Positive Direction \(X\) Axis and Positive Direction \(Y\) Axis are 90\(^\circ\) apart from each other in the Direction of Increase of \(\phi\)) on the \(X-Y\) Plane (which serves as the Equatorial Plane)
      
      \(x=r \sin\theta \cos\phi\hspace{7mm}y=r \sin\theta \sin\phi \hspace{7mm}z=r \cos\theta\)
   2. When the Polar Angle \(\theta\) is measured from the North Pole of the Sphere that lies on Positive Direction of \(Y\) Axis (which serves as the Polar Axis) and the Equatorial Angle \(\phi\) is measured from the Positive Direction of \(Z\) Axis (Such that the Positive Direction \(Z\) Axis and Positive Direction \(X\) Axis are 90\(^\circ\) apart from each other in the Direction of Increase of \(\phi\)) on the \(Z-X\) Plane (which serves as the Equatorial Plane)
      
      \(x=r \sin\theta \sin\phi\hspace{7mm}y=r \cos\theta \hspace{7mm}z=r \sin\theta \cos\phi\)
   3. When the Polar Angle \(\theta\) is measured from the North Pole of the Sphere that lies on Positive Direction of \(X\) Axis (which serves as the Polar Axis) and the Equatorial Angle \(\phi\) is measured from the Positive Direction of \(Y\) Axis (Such that the Positive Direction \(Y\) Axis and Positive Direction \(Z\) Axis are 90\(^\circ\) apart from each other in the Direction of Increase of \(\phi\)) on the \(Y-Z\) Plane (which serves as the Equatorial Plane)
      
      \(x=r \cos\theta\hspace{7mm}y=r \sin\theta \cos\phi\hspace{7mm}z=r \sin\theta \sin\phi\)
6. Conversely any Coordinate Point (\(x,y,z\)) given in Right Handed 3 Dimensional Cartesian Coordinate System can be converted to a Coordinate Point (\(r, \theta, \phi\)) in Spherical Coordinate System in the following ways
   1. When the \(Z\) Axis serves as the Polar Axis and \(X\) and \(Y\) Axis form the Equatorial Plane
      
      \(r=\sqrt{x^2 + y^2 + z^2 }\hspace{7mm}\theta=\arccos({\Large \frac{z}{\sqrt{x^2+y^2+z^2}}})\hspace{7mm}\phi=m\_arctan2\hspace{1mm}(x,y)\)
   2. When the \(Y\) Axis serves as the Polar Axis and \(Z\) and \(X\) Axis form the Equatorial Plane
      
      \(r=\sqrt{x^2 + y^2 + z^2 }\hspace{7mm}\theta=\arccos({\Large \frac{y}{\sqrt{x^2+y^2+z^2}}})\hspace{7mm}\phi=m\_arctan2\hspace{1mm}(z,x)\)
   3. When the \(X\) Axis serves as the Polar Axis and \(Y\) and \(Z\) Axis form the Equatorial Plane
      
      \(r=\sqrt{x^2 + y^2 + z^2 }\hspace{7mm}\theta=\arccos({\Large \frac{x}{\sqrt{x^2+y^2+z^2}}})\hspace{7mm}\phi=m\_arctan2\hspace{1mm}(y,z)\)
   The \(m\_arctan2\) is the Modified Arctangent Function , which calculates the angle \(\phi\) such that \(0 \leq \phi < 2\pi\).