Graph of an Ellipse (Whispering galleries!)
An ellipse is a closed curve which looks like a squashed circle, it is often
described as an oval. It has 2 axes, the longer one is called the major
axis, the shorter one is the minor axis.
The analytical geometry formula for an ellipse is
x2 /a2 + y2/b2 =
1 where a
is
half the length of the major axis and b
is half the length of the minor axis. A circle is really a special
case of an ellipse in which the major and minor axes are equal, so a
= b = r = radius of the circle. The formula for a circle
would then become:-
x2/r2
+ y2/r2 =1
which simplifies to x2
+ y2 = r2
The example plotted here has a major axis of 70
and a minor axis of 40. The formula has been
re-arranged so the y term is on the left side, becoming
y2 = b2 ( 1 - x2/a2)
then simplified to
y2 = b2 - x2b
2/a2
When a is 70
and b is 40,
this becomes
y2 =
1600 - 1600/4900 times x2
which becomes
y2
= 1600 - 0.3265 x2 so our
polynomial factors entered at top of the screen are a=
0, b= -0.3264, c= 0 and d= 1600
Cathedral domes with elliptical shapes become whispering galleries, because the sound waves are reflected off the elliptical sides at shallow angles, thereby being propagated all the way around the ellipse. This can produce amazing acoustic effects when a choir is singing at full throttle underneath such a dome! Examples include St Paul's Cathedral in London and St Peter's Basilica in Rome.