The Formula of a ROTATED Ellipse is:

$$\dfrac {((X-C_x)\cos(\theta)+(Y-C_y)\sin(\theta))^2}{(R_x)^2}+\dfrac{((X-C_x) \sin(\theta)-(Y-C_y) \cos(\theta))^2}{(R_y)^2}=1$$

- $(C_x, C_y)$ 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


1 Answer 1


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:

$$ x(\alpha) = R_x \cos(\alpha) \\ y(\alpha) = R_y \sin(\alpha) $$

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

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

- $C_x$ 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.

  • 2
    $\begingroup$ The generic rotation mapping seems wrong, although the final result ("Once we put the Ellipse equation...") is correct. $\endgroup$
    – polettix
    Jul 9, 2020 at 10:29

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