# 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

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

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

## 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 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.

• The generic rotation mapping seems wrong, although the final result ("Once we put the Ellipse equation...") is correct. Jul 9 '20 at 10:29