does anyone know how to apply a transformation to a unit circle to get an ellipse?

Question

jamesj · Answer

You want to dilate one of the axis.  Suppose you dilate the x-axis.  Then the transformation will be
$$ \left( \begin{array}{cc} \lambda && 0 \ 0 && 1 \end{array} \right) $$

Under that transformation, the unit circle
$$ x^2 + y^2 = 1 $$
becomes
$$ \lambda^2x^2 + y^2 = 1 $$
which is an ellipse.

anonymous · Answer

the transformation i'm trying to apply is to get the ellipse x^2/36+y^2/12=1.. how might i go about doing that?

jamesj · Answer

well, under the transformation I wrote down, the ellipse can also be written as 
$$ \frac{x^2}{(1/\lambda)^2} + y^2 = 1 $$

So what value of lambda would give you the first term in your ellipse?

Now think about how to scale y to get your second term.  

What then does the matrix look like that scales both of those terms appropriately?

anonymous · Answer

soo.. it would look like

(1/36  0 )
(0     1/12) 

?

jamesj · Answer

almost.  Notice we square x and y, and their results in the linear transformation.

jamesj · Answer

got it?

anonymous · Answer

ohh so 1/6 and 1/sqrt12?

jamesj · Answer

Yes

anonymous · Answer

am i allowed to ask another question lol?

jamesj · Answer

Sure, post it on the left.  Protocol now says you give me "Good answer" ;-)