What is the parametric equation of a rotated Ellipse (given the angle of rotation) The Formula of a ROTATED Ellipse is:
$$\dfrac {((X-C_x)\cos(\theta)+(Y-C_y)\sin(\theta))^2}{(R_x)^2}+\dfrac{((X-C_x) \sin(\theta)-(Y-C_y) \cos(\theta))^2}{(R_y)^2}=1$$

There:
- $(C_x, C_y)$ is the center of the Ellipse.
- $R_x$ is the Major-Radius, and $R_y$ is the Minor-Radius.
- $\theta$ is the angle of the Ellipse rotation.

What is the parametric equation of the Ellipse - equations of X and Y - given the Radiuses, Center, Angle to the Point ($\alpha$), and Angle of Ellipses rotation ($\theta$)?? 
See the graph of the rotated ellipse at: https://www.desmos.com/calculator/fu0ko0hali
 A: Step 1 - Parametric Equation of an Ellipse
The parametric formula of an Ellipse - at (0, 0) with the Major Axis parallel to X-Axis and Minor Axis parallel to Y-Axis:
$$
x(\alpha) = R_x \cos(\alpha) \\
y(\alpha) = R_y \sin(\alpha)
$$
Where:

*

*$R_x$ is the major radius

*$R_y$ is the minor radius
Step 2 - Rotate the Equation
To rotate any formula we use the rotation mapping:
$$
x = t \cos(\theta) - f(t) \sin(\theta) \\
y = t \sin(\theta) + f(t) \cos(\theta)
$$
Where:

*

*$\theta$ is the rotation angle

*$t$ is the parameter of the original function

*$f(t)$ is the original function
Once we put the Ellipse equation in the rotation equation we get:
$$
x(\alpha) = R_x \cos(\alpha) \cos(\theta) - R_y \sin(\alpha) \sin(\theta) \\
y(\alpha) = R_y \cos(\alpha) \sin(\theta) + R_x \sin(\alpha) \cos(\theta)
$$
Step 3 - Shift the Equation from the center at (0, 0)
To shift any equation from the center we add $C_x$ to the $x$ equation and $C_y$ to the $y$ equation.
Therefore the equation of a Rotated Ellipse is:
$$
x(\alpha) = R_x \cos(\alpha) \cos(\theta) - R_y \sin(\alpha) \sin(\theta) + C_x \\
y(\alpha) = R_y \cos(\alpha) \sin(\theta) + R_x \sin(\alpha) \cos(\theta) + C_y
$$
Where:

*

*$C_x$ is center X.

*$C_y$ is center Y.

*$R_x$ is the major radius.

*$R_y$ is the minor radius.

*$\alpha$ is  the parameter, which ranges from 0 to 2π radians.

*$\theta$ is the Ellipse rotation angle.
