ELLIPSE MANIA

by Dennis Sparks

Today we are going to look at a family of shapes called ellipses. An ellipse is the set of all points in the plane such that the distances from two given points in the plane , called the foci, is constant. Thus if F and F' are the foci and A and B are any two points on an ellipse, AF + AF'= BF + BF'. CO is a semi-major axis . EO is a semi minor axis. Point O is called the center of the ellipse. Here is an example of an ellipse:

Show[Graphics[{Circle[{0,0},{10,5}],
                    {PointSize[0.010],Point[{2,Sqrt[24]}]},
                    {PointSize[0.010],Point[{-2,Sqrt[24]}]},
                    {PointSize[0.010],Point[{Sqrt[75],0}]},
                    {PointSize[0.010],Point[{-1 Sqrt[75],0}]},
                    Text["F",{Sqrt[75],0.40}],Text["E",{0,5.4}],Text["O",{.4,.4}],
                    Text["F'",{-1 Sqrt[75],0.40}],
                    Text["A",{2,6}],Text["C",{10.3,0}],Text["D",{-10.3,0}],
                    Text["B",{-2,6}]}],
     Axes->Automatic,PlotRange->All,AspectRatio->Automatic];

1.)  First of all I want you to do some measuring. Measure BF', BF , AF' , AF , DF' , F'O (where O is the origin), CF , FO, <DF'B , <BF'F , <AF'F , <AF'D , <DBF' , <F'BF , <F'BC , <DBC, <F'BC , <DBF , <FBC , <DAF' , <F'AF , <FAC , <DAC , <F'AC , <DBA , <F'BA , <BAF , <BAC , <DAF . Measure any other part of the ellipse you choose.
  
  2.)  Write all generalizations(properties) you feel are true about this ellipse.
  
  3.)  Using a pencil , tape , and string make a circle.
  
  4.)  Using a pencil , tape , and string make an ellipse.
  
  5.)  Does your ellipse follow the generalizations you stated in problem 2? Answer in complete sentences.
  
  6.)  How is your ellipse different from your circle ?
       How is your ellipse like your circle ? 
       
  7.)  On your ellipse label your foci, semi-major axis, semi-minor axis, and center.
  
  
  8.) Name any Pythagorean Theorem Relationship you see in your elllipse.
  
  
  9.)Solve this problem invloving a football stadium and ellipses.

Suppose that you are Chief Mathematician for Ornery and Sly Construction Company. Your company has a contract to build a new football stadium for the Fighting Illini in the form of two concentric ellipses, with the field inside the inner ellipse and the seats between the two ellipses. The seats are in the intersection of the graphs of 
 x¤ + 4y¤ æ 100 and 25x¤ + 36y¤ … 3600 where each
unit on the graph represents 10 meters.

a) Draw a graph of the seating area.

Answer (Click the little box at the right.)

Show[Graphics[{
{RGBColor[1,0,0],Circle[{0,0},{10,5}]},
{RGBColor[0,1,0],Circle[{0,0},{12,10}]},
{PointSize[0.05],Point[{0,0}]}}],AspectRatio->Automatic,
PlotRange->All,Axes->Automatic,AxesLabel->{"x","y"}]

b) From your construction handbook you 
            find the area of an elliptical
            region is `ab where a and b are 
            the semi-axes and `=3.14159.
            The Engineering Department
            estimates that each seat occupies .8
            square meter. What is the seating
            capacity?