# Ellipse equation

1. Mar 29, 2014

### alpha25

Hi, does exist an easy way to change the center of circle or a ellipse in polar coordinates?

thanks!

2. Mar 29, 2014

### HallsofIvy

Staff Emeritus
Yes, and it was given in any text book dealing with the conic sections that I have ever seen:
If the equation of an ellipse centered at (0, 0) is
$$\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1$$
then the same ellipse, centered at (a, b) has equation
$$\frac{(x- a)^2}{a^2}+ \frac{(y-b)^2}{b^2}= 1$$

3. Mar 29, 2014

### dextercioby

Now you have to pass from the cartesian eqns that Halls wrote to polar coordinates and you're done.

4. Mar 29, 2014

### alpha25

Yes thanks...but I need it in polar coordinates

5. Mar 29, 2014

### arildno

Insert polar representations for "x" and "y", multiply out parentheses and simplify and redefine variables/constants.
In particular, remember simplifying trig identities, such as, for example:
$$2\sin^{2}\theta=1-\cos(2\theta)$$

6. Mar 29, 2014

### arildno

Now, HallsofIvy made out a special case, with the center with the same values as the lengths of the semi-axes.

You shouldn't make that restriction here (call one of the (a,b)-pairs (c,d)-for example).

To give you the first step on your way, multiplying up and out, we get (with (c,d) centre coordinates):
$$b^{2}r^{2} \cos^{2}\theta+a^{2}r^{2} \sin^{2}\theta-2b^{2}cr \cos\theta-2a^{2} dr\sin\theta=a^{2}b^{2}-c^{2}-d^{2}$$
There would be various ways to simplify this expression further, and redefing independent constants.

One very compact way of doing so would be to transform your equation into the following form:
$$Ar^{2}\cos\gamma+Br\sin\phi=C$$
where the angle "phi" is a phase-shifted version of "gamma"/2 with a fourth constant D to be determined along with A, B and C (gamma being twice the value of "theta")

Last edited: Mar 29, 2014