# 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

• Commented Feb 11, 2018 at 14:14
• Another one stackoverflow.com/questions/4467121/… Commented Feb 11, 2018 at 14:15
• 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 Commented Feb 11, 2018 at 14:52
• Have you seen this math.stackexchange.com/questions/937259/… ? Commented Feb 11, 2018 at 18:53

## 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.
• 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. Commented May 8 at 16:02