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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

Gil Epshtain
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    Maybe of interest https://math.stackexchange.com/questions/426150/what-is-the-general-equation-of-the-ellipse-that-is-not-in-the-origin-and-rotate and https://math.stackexchange.com/questions/941490/whats-the-parametric-equation-for-the-general-form-of-an-ellipse-rotated-by-any – cgiovanardi Feb 11 '18 at 14:14
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    Another one https://stackoverflow.com/questions/4467121/solutions-for-y-for-a-rotated-ellipse – cgiovanardi Feb 11 '18 at 14:15
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    I saw both of these answers but they answer the question what is the generic form of the Ellipse formula. I already know the generic form of the Ellipse formula, but I can't find a way to isolate X and Y – Gil Epshtain Feb 11 '18 at 14:52
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    Have you seen this https://math.stackexchange.com/questions/937259/whats-the-parametric-equation-for-the-general-form-of-an-ellipse-rotated-by-any ? – cgiovanardi Feb 11 '18 at 18:53

1 Answers1

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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.
Alex M.
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Gil Epshtain
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  • Shouldn't $R_x$ and $R_y$ be swapped for $y(\alpha)$? When $\alpha$ is zero, $\cos(\alpha)$ is 1 and $\sin(\alpha)$ is 0. Your equation in step 2 then becomes: $$ x(\alpha) = R_x \cos(\theta) \ y(\alpha) = R_x \sin(\theta) $$ which is not the same as the previous one in step 1. – Ada Melentyeva May 08 '24 at 16:02
  • When $\alpha$ is zero, I think of it like we are at the "beginning" of the parameterization. Usually when we talk about drawing circles and ellipses, we start at the positive x-axis and move counter clockwise. That holds here. Notice if $\theta = 0$, then $x = R_x$ and $y = 0$.

    But when we have a tilt $\theta$, we need to move the starting point. I imagine that the new start position is found by rotating an $R_x$ radial line from the x-axis up counter clockwise, creating a triangle. This triangle has hypotenuse $R_x$ and legs $R_{x}cos(\theta)$ and $R_{x}sin(\theta)$, as you list above.

     – Bcabs4994 Nov 04 '25 at 22:28