



There
are a couple of interesting details about ellipses.






The
orbits of all of the planets moons, space shuttles, asteroids,
comets 


and
anything else that's out there orbiting around 


are
ellipses 





There are
two special points inside an ellipse. 


These are
called focus points (weird name, I know) 





Focus
points always live along the stretched out direction centerline 


The focus
points are the same distance from the center as each other. 


Like
this: 








There is
a trick that will always work on an ellipse. 


Choose
any point on the ellipse line. 


Measure
the distance from each of the focus points. 


Then add
the two measurements together. 


No matter
what two points on the ellipse line you choose, 


you will
always get the same answer! 











As an
ellipse gets stretched out more and more, 


the focus
points move further from the center of the ellipse 


and
closer to the edge. 











Now these
focus points would just be some sort of math trivia 


except
that the orbits of planets and moons and satellites and stuff like
that 


are all
ellipses. 











Now if
the orbiting thing in the picture is the earth, 


and you
want to draw the sun in the picture too 


where
would you put it? 





Right in
the middle? 





NO! 





It would
be at one of the focus points! 











(Actually,
the sun is not centered EXACTLY on a focus point. 


it is a
teeny, tiny bit off. But that is just a technical detail) 





If the
earth and the sun were the right size in this drawing, 


you would
need a microscope to see them. 


Also, the
earth's ellipse is not stretched out nearly this much. 





OK, so
since the focus points have real world value, 


it would
be good if we were able to calculate where they were. 





Here's
the deal. 


Say we
have an ellipse equation like this: 











Hey,
we've seen this one before ... 





It looks
like this: 











So the
focus points are along the stretched out axis. 


In this
case, that's the Y axis. 





The trick
for finding the distance the focus points are from the center 


works
like this: 





Take the
two denominators from the ellipse equation, 


square
them, and find the difference between them. 


Then take
the square root of that. 





That's
the distance from the center to the focus point. 





So, as an
equation: 








So since
this ellipse is centered at (0,0), 


The focus
points are at (0, 8.660) and (0,8.660) 











On the
last page we met this ellipse: 











Where are
the foci? (That's the plural of focus!) 





OK, this
one is stretched in the Y direction, 


and moved
one unit down and two units to the right. 











Since the
whole ellipse is moved two units to the right, 


the line
that the foci are on is moved two units to the right. 


Since the
whole ellipse is moved one unit down, 


the
left/right centerline is also moved one unit down. 


The
center of the ellipse is at (2, 1) 





The
equation for the distance to the focus points 


works out
the same as before ... 











BUT, that
distance is measured from the center point of this ellipse. 





So the
focus points are at (2, 7.660) and (2, 9.660): 











So when
the sign between the X ^{2} and the Y
^{2} term is a
"+" 


We get a
circle or an ellipse. 


What
would we get if the sign was a minus? 





copyright 2005 Bruce Kirkpatrick 
