# Thread: Rotated ellipse equation sought

1. ## Rotated ellipse equation sought

I'm looking for a Cartesian equation for a rotated ellipse.

h is x-koordinate of the center of the ellipse.
k is y-koordinate of the center of the ellipse.
a is the ellipse axis which is parallell to the x-axis when rotation is zero.
b is the ellipse axis which is parallell to the y-axis when rotation is zero.
phi is the rotation angle.

Here is a cartesian equation for a non-rotated ellipse:
(How do I "put rotation phi" into this?)

Code:
(x-h)^2   (y-k)^2
------- + ------- = 1
a^2       b^2

Here is a parametric form of a rotated ellipse:

x = h+a*cos(t)*cos(phi)-b*sin(t)*sin(phi)
y = k+b*sin(t)*cos(phi)+a*cos(t)*sin(phi)

I'd like to have the x and the y in the same equation, and without the parameter t. I think that this means that I want a Cartesian equation for a rotated ellipse, but I'm sorry if I have misunderstood Cartesian here...

Thanks beforehand and a jolly good weekend to ya' all!
Cheers!

2. Is there some standard way of "rotating" a geometric figure in 2D?

Maybe this needs to be in the geometry forum or something?

3. multiplying by this matrix will rotate the graph by the angle theta

$
\begin{bmatrix}
x' \\
y'
\end{bmatrix}
=
\begin{bmatrix}
\cos \theta && -\sin \theta \\
\sin \theta && \cos \theta
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}

$

where x' and y' are the new rotated coordinates
so now suppose that we have the equations of an ellispe

$x=3+3\cos(t)$ and

$y=2+2\sin(t)$

so if we want to rotate the eqations by 45 degrees or $\frac{\pi}{4} radians$ we get..

$
\begin{bmatrix}
\cos \frac{\pi}{4} && -\sin \frac{\pi}{4} \\
\sin \frac{\pi}{4} && \cos \frac{\pi}{4}
\end{bmatrix}
\begin{bmatrix}
3+3\cos(t) \\
2+2\sin(t)
\end{bmatrix}

$

$
\begin{bmatrix}
\frac{\sqrt{2}}{2} && -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} && \frac{\sqrt{2}}{2}
\end{bmatrix}
\begin{bmatrix}
3+3\cos(t) \\
2+2\sin(t)
\end{bmatrix}
$

so then

$x'=\frac{3\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\cos(t)-\sqrt{2}-\sqrt{2}\sin(t)$
and
$y'=\sqrt{2}+\sqrt{2}\sin(t)+ \frac{3\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\cos(t)$

I hope this helps.

4. Actually, you demonstrated how to derive the parametric equations of a rotated ellipse, which I posted, from the parametric equations of a non-rotated ellipse. I can follow the matrix multiplication, yes, but is it possible to put that together in the way I'm looking for:

Joining x and y in one single equation, and getting rid of that t parameter? If I try to substitute and solve from the parametric equation system, I end up with a mess. There must be a much simpler way.

For a non-rotated ellipse, it looks quite simple:

Code:
(x-h)^2   (y-k)^2
------- + ------- = 1
a^2       b^2

How do I turn THAT phi degrees?

5. Hi,
i also had the same problem. I found the solution at the following link. I do not know whether the external links are allowed on this forum or not...

The Quadratic Curves, Circles and Ellipses FAQ

Look for the question ---

Q21. How do I calculate the coefficients of an arbitary rotated ellipse?

I hope this is what you were also looking for .

,

,

### parametric equation of rotated ellipse

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