Finding Polar and Equatorial Angles of a Point

1. Polar and Equatorial Angles are Angles that a Line joining a Point and the Origin makes with the Positive Direction of Polar and Equatorial Axes.
2. The Polar Angle \(\phi\) that any Point \((x,y,z)\) makes with the Positive Direction of Polar Axis can be found using Arccosine Fucntion as follows
   
   

   X=Polar Axis Y=Z=Equatorial Axis	Y=Polar Axis Z=X=Equatorial Axis	Z=Polar Axis X=Y=Equatorial Axis
   \(\phi=\arccos(\frac{x}{\sqrt{x^2+y^2+z^2}})=\alpha\)	\(\phi=\arccos(\frac{y}{\sqrt{x^2+y^2+z^2}})=\beta\)	\(\phi=\arccos(\frac{z}{\sqrt{x^2+y^2+z^2}})=\gamma\)
   The value of the Polar Angle \(\phi\) comes out such that \(0\leq \phi \leq \pi\)

3. The Equatorial Angle \(\theta\) is typically calculated using a function \(arctan2\) which calculates the value of \(\theta\) such that \(-\pi < \theta \leq \pi\). However, calculating Equatorial Angle \(\theta\) such that \(0\leq \theta < 2\pi\) is more useful, which can be done by using the formulae for Modified Arctangent Function \(m\_arctan2\) as given below
   
   

   For X-Y Plane (X is Horizontal Axis and Y is Vertical Axis)
   \(m\_arctan2\) Function for Counter Clockwize from Positive X Axis via ArcCosine	\(m\_arctan2\) Function for Counter Clockwize from Positive X Axis via ArcSine	\(m\_arctan2\) Function for Clockwize from Positive Y Axis via ArcCosine	\(m\_arctan2\) Function for Clockwize from Positive Y Axis via ArcSine
   if \(x=0\) And \(y=0\) \(\theta=0\) else if \(y\geq0\) \(\theta=\arccos(\frac{x}{\sqrt{x^2+y^2}}) \mod 2\pi\) else if \(y<0\) \(\theta=(\arccos(\frac{-x}{\sqrt{x^2+y^2}}) + \pi) \mod 2\pi\)	if \(x=0\) And \(y=0\) \(\theta=0\) else if \(x < 0\) \(\theta=(\pi - \arcsin(\frac{y}{\sqrt{x^2+y^2}})) \mod 2\pi\) else if \(y \geq 0\) \(\theta=\arcsin(\frac{y}{\sqrt{x^2+y^2}}) \mod 2\pi\) else if \(y < 0\) \(\theta=(\arcsin(\frac{y}{\sqrt{x^2+y^2}}) + 2\pi)\mod 2\pi\)	if \(x=0\) And \(y=0\) \(\theta=0\) else if \(x\geq0\) \(\theta=\arccos(\frac{y}{\sqrt{x^2+y^2}})\mod 2\pi\) else if \(x<0\) \(\theta=(\arccos(\frac{-y}{\sqrt{x^2+y^2}}) + \pi)\mod 2\pi\)	if \(x=0\) And \(y=0\) \(\theta=0\) else if \(y < 0\) \(\theta=(\pi - \arcsin(\frac{x}{\sqrt{x^2+y^2}}))\mod 2\pi\) else if \(x \geq 0\) \(\theta=\arcsin(\frac{x}{\sqrt{x^2+y^2}})\mod 2\pi\) else if \(x < 0\) \(\theta=(\arcsin(\frac{x}{\sqrt{x^2+y^2}}) + 2\pi)\mod 2\pi\)

   
   

   For Y-Z Plane (Y is Horizontal Axis and Z is Vertical Axis)
   \(m\_arctan2\) Function for Counter Clockwize from Positive Y Axis via ArcCosine	\(m\_arctan2\) Function for Counter Clockwize from Positive Y Axis via ArcSine	\(m\_arctan2\) Function for Clockwize from Positive Z Axis via ArcCosine	\(m\_arctan2\) Function for Clockwize from Positive Z Axis via ArcSine
   if \(y=0\) And \(z=0\) \(\theta=0\) else if \(z\geq0\) \(\theta=\arccos(\frac{y}{\sqrt{y^2+z^2}}) \mod 2\pi\) else (if \(z<0\) \(\theta=(\arccos(\frac{-y}{\sqrt{y^2+z^2}}) + \pi) \mod 2\pi\)	if \(y=0\) And \(z=0\) \(\theta=0\) else if \(y < 0\) \(\theta=(\pi - \arcsin(\frac{z}{\sqrt{y^2+z^2}})) \mod 2\pi\) else if \(z \geq 0\) \(\theta=\arcsin(\frac{z}{\sqrt{y^2+z^2}}) \mod 2\pi\) else if \(z < 0\) \(\theta=(\arcsin(\frac{z}{\sqrt{y^2+z^2}}) + 2\pi) \mod 2\pi\)	if \(y=0\) And \(z=0\) \(\theta=0\) else if \(y\geq0\) \(\theta=\arccos(\frac{z}{\sqrt{y^2+z^2}}) \mod 2\pi\) else if \(y<0\) \(\theta=(\arccos(\frac{-z}{\sqrt{y^2+z^2}}) + \pi) \mod 2\pi\)	if \(y=0\) And \(z=0\) \(\theta=0\) else if \(z < 0\) \(\theta=(\pi - \arcsin(\frac{y}{\sqrt{x^2+z^2}})) \mod 2\pi\) else if \(y \geq 0\) \(\theta=\arcsin(\frac{y}{\sqrt{y^2+z^2}}) \mod 2\pi\) else if \(y < 0\) \(\theta=(\arcsin(\frac{y}{\sqrt{y^2+z^2}}) + 2\pi) \mod 2\pi\)

   
   

   For Z-X Plane (Z is Horizontal Axis and X is Vertical Axis)
   \(m\_arctan2\) Function for Counter Clockwize from Positive Z Axis via ArcCosine	\(m\_arctan2\) Function for Counter Clockwize from Positive Z Axis via ArcSine	\(m\_arctan2\) Function for Clockwize from Positive X Axis via ArcCosine	\(m\_arctan2\) Function for Clockwize from Positive X Axis via ArcSine
   if \(x=0\) And \(z=0\) \(\theta=0\) else if \(x\geq0\) \(\theta=\arccos(\frac{z}{\sqrt{x^2+z^2}}) \mod 2\pi\) else if \(x<0\) \(\theta=(\arccos(\frac{-z}{\sqrt{x^2+z^2}}) + \pi) \mod 2\pi\)	if \(x=0\) And \(z=0\) \(\theta=0\) else if \(z < 0\) \(\theta=(\pi - \arcsin(\frac{x}{\sqrt{x^2+z^2}})) \mod 2\pi\) else if \(x \geq 0\) \(\theta=\arcsin(\frac{x}{\sqrt{x^2+z^2}}) \mod 2\pi\) else if \(x < 0\) \(\theta=(\arcsin(\frac{x}{\sqrt{x^2+z^2}}) + 2\pi) \mod 2\pi\)	if \(x=0\) And \(z=0\) \(\theta=0\) else if \(z\geq0\) \(\theta=\arccos(\frac{x}{\sqrt{x^2+z^2}}) \mod 2\pi\) else if \(z<0\) \(\theta=(\arccos(\frac{-x}{\sqrt{x^2+z^2}}) + \pi) \mod 2\pi\)	if \(x=0\) And \(z=0\) \(\theta=0\) else if \(x < 0\) \(\theta=(\pi - \arcsin(\frac{z}{\sqrt{x^2+z^2}})) \mod 2\pi\) else if \(z \geq 0\) \(\theta=\arcsin(\frac{z}{\sqrt{x^2+z^2}}) \mod 2\pi\) else if \(z < 0\) \(\theta=(\arcsin(\frac{z}{\sqrt{x^2+z^2}}) + 2\pi) \mod 2\pi\)

   
   Please note that calculations for Equatorial Angles given in table above is for Right Handed Cartesian Coordinte System. For Left Handed Cartesian Coordinte System, the Horizontal and Vertical Axis get Interchanged which Interchanges the Counter-Clockwize and the Clockwize Angles.