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.


Elliptical.htm     02 November 2006         MENU  Theory -> Maths and Physics -> Mathematical equations for music