Finding Parametric Equations for Axis Aligned and Rotated Hyperbola Based on Hypebolic Sine and Cosine

1. The Parametric Equation of any Hyperbola is given in terms of Length of its Semi-Transverse Axis \(a\), Length of its Semi-Conjugate Axis \(b\) and the Hyperbolic Angle \(\theta\) that the Line joining the Origin and any Point on Hyperbola makes with the Positive Direction of \(X\)-Axis.
2. The Parametric Equation of Axis Aligned Hyperbolas having their Coordinates of Center at Origin are given as
   
   \(x=\pm a\cosh \theta,\hspace{.5cm}y=b\sinh \theta\)   (\(X\)-Transverse Hyperbola)...(1)
   
   \(x=b\sinh \theta,\hspace{.5cm}y=\pm a\cosh \theta\)   (\(Y\)-Transverse Hyperbola)...(2)
   
   The following give the Parametric Equations for Conjugate Hyperbolas for the Hyperbolas given in above equations (1) and (2)
   
   \(x=a\sinh \theta,\hspace{.5cm}y=\pm b\cosh \theta\)   (\(Y\)-Transverse Hyperbola which is Conjugate Hyperbola for equation (1))...(3)
   
   \(x=\pm b\cosh \theta,\hspace{.5cm}y=a\sinh \theta\)   (\(X\)-Transverse Hyperbola which is Conjugate Hyperbola for equation (2))...(4)
3. The Parametric Equation of Hyperbolas Rotated by an Angle \(\phi\) having their Coordinates of Center at Origin are given as
   
   \(x=\pm a\cosh \theta \cos \phi-b\sinh \theta \sin \phi,\hspace{.5cm}y=b\sinh \theta \cos\phi + (\pm a\cosh \theta\sin\phi)\)   ...(5)
   
   The following gives the Parametric Equation for Conjugate Hyperbola for the Hyperbola given in above equation (5)
   
   \(x=a\sinh \theta \cos \phi-(\pm b\cosh \theta \sin \phi),\hspace{.5cm}y=\pm b\cosh \theta \cos\phi + a\sinh \theta\sin\phi\)   ...(6)
4. The Parametric Equation of Axis Aligned Hyperbolas having their Coordinates of Center at \((x_c,y_c)\) are given as
   
   \(x=x_c + \pm a\cosh \theta,\hspace{.5cm}y=y_c + b\sinh \theta\)   (\(X\)-Transverse Hyperbola)...(7)
   
   \(x=x_c + b\sinh \theta,\hspace{.5cm}y=y_c + \pm a\cosh \theta\)   (\(Y\)-Transverse Hyperbola)...(8)
   
   The following give the Parametric Equations for Conjugate Hyperbolas for the Hyperbolas given in above equations (7) and (8)
   
   \(x=x_c + a\sinh \theta,\hspace{.5cm}y=y_c + \pm b\cosh \theta\)   (\(Y\)-Transverse Hyperbola which is Conjugate Hyperbola for equation (7))...(9)
   
   \(x=x_c + \pm b\cosh \theta,\hspace{.5cm}y=y_c + a\sinh \theta\)   (\(X\)-Transverse Hyperbola which is Conjugate Hyperbola for equation (8))...(10)
5. The Parametric Equation of Hyperbolas Rotated by an Angle \(\phi\) having their Coordinates of Center at \((x_c,y_c)\) are given as
   
   \(x=x_c + (\pm a\cosh \theta \cos \phi-b\sinh \theta \sin \phi),\hspace{.5cm}y=y_c + (b\sinh \theta \cos\phi + \pm a\cosh \theta\sin\phi)\)   ...(11)
   
   The following gives the Parametric Equation for Conjugate Hyperbola for the Hyperbola given in above equation (11)
   
   \(x=x_c + (a\sinh \theta \cos \phi- \pm b\cosh \theta \sin \phi),\hspace{.5cm}y=y_c + (\pm b\cosh \theta \cos\phi + a\sinh \theta\sin\phi)\)   ...(12)