Finding Parametric Equations for Axis Aligned and Rotated Ellipse

1. The Parametric Equation of any Ellipse is given in terms of Length of its Semi-Major Axis \(a\), Length of its Semi-Minor Axis \(b\) and the Angle \(\theta\) that any Point on Ellipse makes with the Positive Direction of \(X\)-Axis.
2. The Parametric Equation of Axis Aligned Ellipses having their Coordinates of Center at Origin are given as
   
   \(x=a\cos \theta,\hspace{.5cm}y=b\sin \theta\)   (\(X\)-Major Ellipse)...(1)
   
   \(x=b\cos \theta,\hspace{.5cm}y=a\sin \theta\)   (\(Y\)-Major Ellipse)...(2)
3. The Parametric Equation of Ellipses Rotated by an Angle \(\phi\) having their Coordinates of Center at Origin are given as
   
   \(x=a\cos \theta \cos \phi-b\sin \theta \sin \phi,\hspace{.5cm}y=b\sin \theta \cos\phi + a\cos \theta\sin\phi\)   ...(3)
   
   
4. The Parametric Equation of Axis Aligned Ellipses having their Coordinates of Center at \((x_c,y_c)\) are given as
   
   \(x=x_c + a\cos \theta,\hspace{.5cm}y=y_c + b\sin \theta\)   (\(X\)-Major Ellipse)...(4)
   
   \(x=x_c + b\cos \theta,\hspace{.5cm}y=y_c + a\sin \theta\)   (\(Y\)-Major Ellipse)...(5)
5. The Parametric Equation of Ellipses Rotated by an Angle \(\phi\) having their Coordinates of Center at \((x_c,y_c)\) are given as
   
   \(x=x_c + (a\cos \theta \cos \phi-b\sin \theta \sin \phi),\hspace{.5cm}y=y_c + (b\sin \theta \cos\phi + a\cos \theta\sin\phi)\)   ...(6)