Constructing Axes of an Ellipse with Straightedge and Compass

Date: 01/23/2006 at 15:41:53
From: philip
Subject: I want to find the centerline of an ellipse

If you encounter an ellipse in a CAD drawing or on a blackboard, how 
can you determine the centerline?  Either the short or long axis 
would be acceptable. 

I think there must be a way to construct these lines working just from
the ellipse itself, using straight edge and compass.  I have an idea
how this could be done, however, I don't have the math skills to prove
my method.  My definition of an ellipse is the shape you get from
sectioning a cylinder at an angle other than 90º to the cylinder's axis.

Date: 01/25/2006 at 13:51:02
From: Doctor Wilkinson
Subject: Re: I want to find the centerline of an ellipse

Hi, Philip.  Thanks for the interesting problem!  The first step is to
find the center of the ellipse.  To do this we use the fact that the
line through the midpoints of two parallel chords of the ellipse
passes through the center.  So take any chord.  Take another point on
the ellipse and construct a parallel chord.  Then bisect the chords
and draw a line through the midpoints.  Do the same for another chord
and you will have the center.

Once you have the center, just draw a circle intersecting the ellipse.
This gives you four points which will be the vertices of a rectangle
and you can now easily construct the axes of the ellipse.  For 
example, join the opposite corners and take the angle bisectors of the 
angles formed at the center.

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