How to Parameterize an Ellipse with Offsets?

[tysonk]

How do I parameterize the following?
x^{2}/a^{2} + y^{2}/b^{2} -2x/a -2y/b = 0

I tried letting x =t or some other parameters but found it difficult to solve for y.

 

[fzero]

Try to write

\frac{x^2}{a^2} - \frac{2x}{a} =\left( \frac{x}{a} -c \right)^2 +d

for some c,d to be determined. Do the same thing with the y terms. This is called completing the square.

 

[tysonk]

Thanks!
So letting t=x I was able to get a parameterized equation for y and x.
Is there any obvious thing I should let x be. Or can I just set it to whatever and solve y in terms of that?

 

[fzero]

Thanks!
So letting t=x I was able to get a parameterized equation for y and x.
Is there any obvious thing I should let x be. Or can I just set it to whatever and solve y in terms of that?

Do you know what shape is described by your equation? When you figure it out, you might find a nice parametrization in terms of trig functions.