Rules for Measurement of Rotation Angles on a Plane

1. Rotation Angles on a Plane are measured with reference to a Set of Mutually Perpendicular Horizontal and Vertical Axes similar to that of a 2 Dimensional Cartesian Coordinate System.
2. Rotation Angles on a Plane are typically measured in 4 following ways
   1. Counter-Clockwize Rotation from Positive Side of Horizontal Axis (which is same as Positive \(X\) Axis in 2D Cartesian Coordinate System).
   2. Counter-Clockwize Rotation from Positive Side of Vertical Axis (which is same as Positive \(Y\) Axis in 2D Cartesian Coordinate System).
   3. Clockwize Rotation from Positive Side of Horizontal Axis .
   4. Clockwize Rotation from Positive Side of Vertical Axis.
3. Counter-Clockwize Rotations are considered to have Positive Rotation Angles. Clockwize Rotations are considered to have Negative Rotation Angles.
4. Rotation Angle Measurements follow a Modulo 360 arithmetic, i.e. Rotation Angle Measurements wrap around to Zero on reaching \(\pm 360\)° (or 2π Radians).
5. Objects with a Rotational Symmetry of \(\phi\) follow a a Modulo \(\phi\) arithmetic, i.e. Rotation Angle Measurements for such Objects wrap around to Zero on reaching \(\pm \phi\) (Degrees or Radians)
6. If Counter-Clockwize Rotation Angle \(\theta\) from Positive Side of Horizontal Axis is known, then Counter-Clockwize Rotation Angle \(\sigma\) from the Positive Side of Vertical Axis can be calculated as
   
   \(\sigma= (\theta + 270^\circ) \mod 360^\circ = (\theta + \frac{3\pi}{2}) \mod 2\pi \)
   
   Similarly, if Counter-Clockwize Rotation Angle \(\sigma\) from Positive Side of Vertical Axis is known, then Counter-Clockwize Rotation Angle \(\theta\) from the Positive Side of Horizontal Axis can be calculated as
   
   \(\theta= (\sigma + 90^\circ) \mod 360^\circ = (\sigma + \frac{\pi}{2}) \mod 2\pi \)
7. If Clockwize Rotation Angle \(\theta\) from Positive Side of Horizontal Axis is known, then Clockwize Rotation Angle \(\sigma\) from the Positive Side of Vertical Axis can be calculated as
   
   \(\sigma= -((|\theta| + 90^\circ) \mod 360^\circ) = -((|\theta| + \frac{\pi}{2}) \mod 2\pi) \)
   
   Similarly, if Clockwize Rotation Angle \(\sigma\) from Positive Side of Vertical Axis is known, then Clockwize Rotation Angle \(\theta\) from the Positive Side of Horizontal Axis can be calculated as
   
   \(\theta= -((|\sigma| + 270^\circ) \mod 360^\circ) = -((|\sigma| + \frac{3\pi}{2}) \mod 2\pi) \)
8. If Counter-Clockwize Rotation Angle \(\theta\) from the Positive Side of Horizontal Axis is known, then corresponding Clockwize Rotation Angle \(\sigma\) from the Positive Side of Vertical Axis can be calculated as
   
   \(\sigma= ((360^\circ - |\theta|) + 90^\circ) \mod 360^\circ =((2\pi - |\theta|) + \frac{\pi}{2}) \mod 2\pi\)
   
   Similarly, If Clockwize Rotation Angle \(\theta\) from the Positive Side of Vertical Axis is known, then corresponding Counter-Clockwize Rotation Angle \(\sigma\) from the Positive Side of Horizontal Axis can be calculated using the above formula.
   
   Please note that \(\sigma\) is Negative for Clockwize Angle and Positive for Counter Clockwize Angle.
9. If Clockwize Rotation Angle \(\theta\) from the Positive Side of Horizontal Axis is known, then corresponding Counter-Clockwize Rotation Angle \(\sigma\) from the Positive Side of Vertical Axis can be calculated as
   
   \(\sigma= ((360^\circ - |\theta|) + 270^\circ) \mod 360^\circ= ((2\pi - |\theta|) + \frac{3\pi}{2}) \mod 2\pi\)
   
   Similarly, If Counter-Clockwize Rotation Angle \(\theta\) from the Positive Side of Vertical Axis is known, then corresponding Clockwize Rotation Angle \(\sigma\) from the Positive Side of Horizontal Axis can be calculated using the above formula.
   
   Please note that \(\sigma\) is Negative for Clockwize Angle and Positive for Counter Clockwize Angle.
10. Every Counter-Clockwise Rotation Angle has a corresponding Clockwise Rotation Angle and vice versa. The following table illustrates the formulae for calculating the same with examples
    
    

    Rotation Type/Angle	Corresponding Rotation Type/Angle	Example
    Counter-Clockwise by \(\theta\)	Clockwise by \((\theta \bmod 360) - 360\)	If Counter-Clockwise = 732° \(\Rightarrow\) Actual Counter-Clockwise=\((732 \bmod 360)\)=12° \(\Rightarrow\) Clockwise = 12° - 360° = -348°
    Clockwise by \(\theta\)	Counter-Clockwise by \((\theta \bmod 360) + 360\)	If Clockwise = -732° \(\Rightarrow\) Actual Clockwise=\((-732 \bmod 360)\)=-12° \(\Rightarrow\) Counter-Clockwise = -12° + 360° = 348°

    
    The formule given in the table above can be used for Objects with a Rotational Symmetry of \(\phi\) by just replacing 360 with \(\phi\).
11. Every Rotation Angle \(\theta\) has a Diametrically Opposite Rotation Angle \(\sigma\) given by
    
    \(\sigma= (|\theta| + 180\)°) \(\mod 360\)°
    
    Please note that both \(\theta\) and \(\sigma\) are Positive for Counter-Clockwize Rotation Angles and Negative for Clockwize Rotation Angles
12. If \(\theta\) is a Counter-Clockwize Positive Rotation Angle \(\leq 180\)° then Diametrically Opposite Clockwize Negative Rotation Angle is given by \(\theta-180\)°.
    
    If \(\theta\) is a Clockwize Negative Rotation Angle \(\geq -180\)° then Diametrically Opposite Counter-Clockwize Positive Rotation Angle is given by \(\theta+180\)°
13. The value of ArcCosine Function \(\theta\) for Cosine Value of any given Angle always gets calculated between \(0\) and \(\pi\). However the Actual Value of the Angle can be either \(\theta\) or \(2\pi - \theta\).
14. The value of ArcSine Function \(\theta\) for Sine Value of any given Angle always gets calculated between \({\Large -\frac{\pi}{2}}\) and \({\Large \frac{\pi}{2}}\). However if \(\theta > 0 \) then the Actual Value of the Angle can be either \(\theta\) or \(\pi - \theta\). And if \(\theta < 0 \) then the Actual Value of the Angle can be either \(\theta\) or \(-\pi - \theta\).