



So the
equation of a circle looks something like:






X^{2}
+ Y^{2} = 5^{2}






If the
numbers next to the X ^{2} and Y
^{2} are different
from each other, 


the
circle gets stretched ... 








To see
how this stretching works, 


let's
start with: 


4X^{2}
+ Y^{2} = 100






The
coefficient on the X ^{2} (4) 


is
different from the coefficient on the Y
^{2} (1) 


and both
terms are positive. 





That
means we have an ellipse 





But we
can't tell yet how big it is, 


or which
direction it is stretched. 





To work
these out, 


first
divide both sides by the number on the right side 


and
simplify ... 








The main
point is to get the right side of the equation to be 1. 


Next, we
need to get the left side of the equation to look like: 











To do
that, we need to figure out the square roots 


of the
numbers in the denominators. 





Sometimes
these work out to be whole numbers 


sometimes
they don't. 


To keep
things a bit cleaner in this example, they do ... 











So
wonderful, 


now that
we've gone through all of these shenanigans, 


what have
we got? 





Well now
we know that the ellipse goes to + 5 and  5 in the X direction, 


and out
to + 10 and  10 in the Y direction. 











OH YEAH? 


Just how
do we know that? 





We know
that because those are the X and Y term denominators. 











When X is
zero the whole first term on the left side is zero. 


The only
way to make the left side equal to 1 


is for Y
to equal 10 or 10. 





When Y is
zero the whole second term on the left side is zero. 


The only
way to make the left side equal to 1 


is for X
to equal 5 or 5. 





That
ellipse was centered at the origin (0,0). 


Moving
the ellipse center away from the origin 


works
just like it did with a circle. 





Subtract
something from X moves it that far right 


Add
something to X moves it that far left. 


Subtract
something from Y moves it that far up. 


Add
something to Y moves it that far down. 





Example: 


Say you
wanted the equation 


of an
ellipse that was the same size as the one in the last example 


and
centered at X = 2, Y  1. 











copyright 2005 Bruce Kirkpatrick 
