Graphing an Ellipse

Date: 11/20/98 at 20:23:50
From: Phil
Subject: Coordinate geometry

A family friend was helping me with my math homework, and mentioned 
that once, while she was working with coordinate geometry in her senior 
year, she graphed a "really nice" ellipse.  She says she forgot how, 
and my math teacher says I won't get to that until a lot later. How do 
you graph an ellipse? What is the equation? Is it really coordinate 
geometry, or does it just involve coordinates?

Date: 01/08/99 at 13:28:01
From: Doctor Ujjwal
Subject: Re: Coordinate geometry

Dear Phil,

I am glad that you are attracted by the beauty of geometric shapes. 
Indeed there are many ways of constructing nice ellipses. The equipment 
used may vary from two pins and a string to computers. But since you 
are trying to graph one, let's see a graphical method. Remember that 
this is just one of the many, many constructions.

A very straightforward method is to use one of the many forms of the 
equation of an ellipse. For example:

   x^2   y^2
   --- + --- = 1
   a^2   b^2

where a and b are constants. The greater of the two is called the 
semi-major axis and the other is the semi-minor axis. Values of (x, y) 
can be calculated from such an equation, plotted on graph paper, and 
then connected to get an ellipse, but this is a tedious method so let's 
leave it to computers.

There is a fun way of plotting an ellipse. It also brings out the 
essential nature of the ellipse as a circle 'stretched' in one 
direction.
 
Draw the horizontal (x) axis and vertical (y) axes on graph paper. The 
point (O) where the two axes meet is called the origin. With a compass 
draw a circle centered at the origin. Next we are going to stretch this 
circle in the x direction. Read the coordinates (x, y) of any point (C) 
on the circle. Plot a new point (E) with coordinates (2*x, y). Point 
(E) is on an ellipse we would get by stretching the circle to double 
its size in the x direction.

    ^ Y-axis                
    |           C(x,y)      E(2*x,y) 
   y|<--- x --->*<--- x --->*          
    |                        
    |                        
   -|------------------------------> X-axis      

Repeat it for more points on the circle and you will see an ellipse 
taking shape.

If you get tired of calculations, you can use the divider in your 
compass box to mark the x coordinate of (C) and mark it horizontally 
from (C) to get point (E). You can also save a lot of time by using the 
symmetry of an ellipse about the X and Y axes. 

                     ^ Y axis                
    E1               |                E 
    *<----- x ------>|<----- x ------>*          
    |                |                |
    y                |                y
    |                |                |
   ------------------|------------------------------> X axis      
    |                |                |
    y                |                y
    |                |                | 
    *<----- x ------>|<----- x ------>*          
    E3               |                E2

Can you figure out how you can reflect the point (E) about the X and Y 
axes to get three more points (E1, E2, E3) on the ellipse? Here too the 
divider comes in very handy.

Good luck and happy plotting!

- Doctor Ujjwal Rane, The Math Forum
  http://mathforum.org/dr.math/