Accurate Drawing of an Ellipse

Date: 02/14/99 at 13:43:32
From: Patrick
Subject: Accurate Drawing of an Ellipse

I am an architectural student and am designing a building with an 
ellipse shape. My problem is that I have forgotten how to draw it on my 
drafting table. I know an ellipse can be drawn using 2 fixed pins tied 
with a thread, but this method is not accurate enough. So my question 
is, having my 2 foci fixed, how can I draw an ellipse accurately using 
simple tools?

Thanks.

Date: 02/16/99 at 08:53:11
From: Doctor Peterson
Subject: Re: Accurate Drawing of an Ellipse

I think what you're looking for is a way to plot an ellipse using 
standard drafting tools without the error introduced by the thickness 
of the string or by varying tension in it. There are several ways to 
plot points of an ellipse, which you can then fill in to make the 
curve. Probably the easiest is the method using two circles. For this 
you need to know the minor and major axes; I'm not sure what you're 
given to start with, but even if it is the foci and the string length, 
you can figure out the axes from that. (The major axis is the string 
length, and the minor axis can be found using the Pythagorean theorem 
from that and the distance between foci.)

Here's an attempt to draw the method. Draw two concentric circles whose 
diameters are the major and minor axes of the desired ellipse. To plot 
a point of the ellipse, draw a radius of the larger circle, and draw a 
line vertically from the large one and a line horizontally from the 
small one. Where these intersect is a point on the ellipse. Draw enough 
of those and you can make an accurate curve.

                       ***********     |/
                 ******           *****+
              ***                     /|***
           ***                       / |   ***
          *                         /  |      *
         *                         /   |       *
       **          .....*********./... |        **
      *      ......  ***        -+**---+-...      *
     *    ...      **           /   ** |    ...    *
     * ...        *            /      *        ... *
    *..          *            /        *          ..*
    *           *            /          *           *
    *           *           +           *           *
    *           *                       *           *
    *..          *                     *          ..*
     * ...        *                   *        ... *
     *    ...      **               **      ...    *
      *      ......  ***         ***  ......      *
       **          .....*********.....          **
         *                                     *
          **                                 **
            **                             **
              ***                       ***
                 *******         *******
                        *********

Let me know if that doesn't meet your needs for some reason.

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