Register to reply

Ellipse equation

by alpha25
Tags: ellipse, equation
Share this thread:
Mar29-14, 09:49 AM
P: 9
Hi, does exist an easy way to change the center of circle or a ellipse in polar coordinates?

Phys.Org News Partner Mathematics news on
Heat distributions help researchers to understand curved space
Professor quantifies how 'one thing leads to another'
Team announces construction of a formal computer-verified proof of the Kepler conjecture
Mar29-14, 02:49 PM
Sci Advisor
PF Gold
P: 39,544
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
[tex]\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1[/tex]
then the same ellipse, centered at (a, b) has equation
[tex]\frac{(x- a)^2}{a^2}+ \frac{(y-b)^2}{b^2}= 1[/tex]
Mar29-14, 03:07 PM
Sci Advisor
HW Helper
P: 11,926
Now you have to pass from the cartesian eqns that Halls wrote to polar coordinates and you're done.

Mar29-14, 04:24 PM
P: 9
Ellipse equation

Yes thanks...but I need it in polar coordinates
Mar29-14, 04:34 PM
Sci Advisor
HW Helper
PF Gold
P: 12,016
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:
Mar29-14, 06:12 PM
Sci Advisor
HW Helper
PF Gold
P: 12,016
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):
[tex]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}[/tex]
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:
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")

Register to reply

Related Discussions
How can i know that this equation is a ellipse? General Math 4
Need help on ellipse equation General Math 5
Equation of an Ellipse. Precalculus Mathematics Homework 3
Equation for an ellipse Precalculus Mathematics Homework 8
Equation of ellipse in 3D General Math 11