What is the parametric equation of a rotated Ellipse (given the angle of rotation)

Question

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

Answer 1

Step 1 - The parametric equation of an ellipse

The parametric formula of an ellipse centered at $(0, 0)$, with the major axis parallel to the $x$-axis and minor axis parallel to the $y$-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

The rotation is given by the mapping: $$ x(\theta) = \cos(\theta) x - \sin(\theta) y\\ y(\theta) = \sin(\theta) x + \cos(\theta) y $$ where:

• $\theta$ is the rotation angle.

Once we put the ellipse equation in the rotation equation we get: $$ x(\alpha) = R_x \cos(\alpha) \cos(\theta) - R_y \sin(\alpha) \sin(\theta) \\ y(\alpha) = R_x \cos(\alpha) \sin(\theta) + R_y \sin(\alpha) \cos(\theta) $$

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: $$ x(\alpha) = R_x \cos(\alpha) \cos(\theta) - R_y \sin(\alpha) \sin(\theta) + C_x \\ y(\alpha) = R_x \cos(\alpha) \sin(\theta) + R_y \sin(\alpha) \cos(\theta) + C_y $$

where:

• $C_x$ is the $x$-coordinate of the center
• $C_y$ is the $y$-coordinate of the center
• $R_x$ is the major radius
• $R_y$ is the minor radius
• $\alpha$ is the parameter, which ranges from $0$ to $2 \pi$ radians
• $\theta$ is the ellipse rotation angle.