Drawing an Ellipse Using Conjugate Diameters

Date: 12/07/2005 at 12:31:52
From: Jason
Subject: Drawing an ellipse using Conjugate Diameters

I am currently working to implement a CAD-type file format for a
computer program at work.  In order to fully implement this format, I
will need to be able to draw an ellipse or elliptical arc with the
information provided in the file.  There are a few problems with this
situation, but the most important is pulling enough information from
the supplied data to fully map the ellipse in terms that the drawing
library can use.

Here is what I am given with the CAD file:

   CenterX - The center X of the ellipse
   CenterY - The center Y of the ellipse
   End1X   - X for the starting point
   End1Y   - Y for the starting point
   End2X   - X for the end point (clockwise)
   End2Y   - Y for the end point (clockwise)

In order to draw an ellipse, I have to pass the bounding rectangle,
starting angle and the angle between the starting and ending points. 
I can rotate drawings by specifying a transformation matrix, provided
I can determine the angle that the ellipse should be rotated.

With only a center point and the starting and ending points, how can I
extract the information needed to draw this ellipse in terms of
bounding rectangle and starting and ending angles?

The whitepapers for this file format mention using Conjugate Diameters
to draw the ellipse, but I'm not sure how to approach this method with
only the three supplied points.

Date: 12/07/2005 at 14:32:31
From: Doctor George
Subject: Re: Drawing an ellipse using Conjugate Diameters


Hi Jason,

Thanks for writing to Doctor Math.

There is quite a bit to this problem. First we need to clarify your
inputs. Conjugate diameters are related to bounding parallelograms. Do
you have a bounding rectangle, or a bounding parallelogram?

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

Date: 12/08/2005 at 09:27:18
From: Jason
Subject: Drawing an ellipse using Conjugate Diameters

I have enough information to create a bounding rectangle.  All of the
information pulled from an Elliptical Arc Input block in this file is
as follows:

  CenterX, 
  CenterY, 
  End1X, 
  End1X, 
  End2X, 
  End2Y, 
  StartVectorX, 
  StartVectorY, 
  EndVectorX, 
  EndVectorY

This creates the three points Center, End1, and End2; and StartVector
and EndVector (with "End1" and "End2" being the major and minor axes). 
It's the End1 and End2 points that I use to determine if the ellipse
should be rotated.

I appreciate the time you've taken to respond to my question.  If you
can help me solve this delima, then you will have saved me a huge
headache.  I must say that the Dr. Math forums have saved me on more
than one occasion, but this is by far the most complex problem I've
had to deal with since finding the site.  Again, thank you for your help.

Date: 12/09/2005 at 07:35:20
From: Doctor George
Subject: Re: Drawing an ellipse using Conjugate Diameters


As Jason wrote to Dr. Math
On 12/08/2005 at 09:27:18 (Eastern Time),
>I have enough information to create a bounding rectangle.  All of the
>information pulled from an Elliptical Arc Input block in this file is
>as follows:
>
>CenterX, CenterY, End1X, End1X, End2X, End2Y, StartVectorX,
>StartVectorY, EndVectorX, EndVectorY
>
>This creates the three points Center, End1, End2, StartVector, and
>EndVector (with "End1" and "End2" being the major and minor axes). 
>It's the End1 and End2 points that I use to determine if the ellipse
>should be rotated.
>
>I appreciate the time you've taken to respond to my question.  If you
>can help me solve this delima, then you will have saved me a huge
>headache.  I must say that the Dr. Math forums have saved me on more
>than one occasion, but this is by far the most complex problem I've
>had to deal with since finding the site.  Again, thank you for your help.

Hi Jason,

If you translate your data so that the center point moves to the
origin the equation of the ellipse can be written as

    ax^2 + bxy + cy^2 = 1

If you substitute one endpoint from each chord into this equation you
will get two linear equations with variables a, b, and c.

Now we need a third equation. Implicitly differentiate the equation of
the ellipse.

    2ax + b(y + x(dy/dx)) + 2cy(dy/dx) = 0

If you substitute one endpoint from a chord and the slope of its
tangent line (slope = dy/dx) into this equation you will get a third
linear equation with variables a, b, and c. Now you just need to solve
the linear system.

As a side note, this method would also work with conjugate diameter
pairs rather than a bounding reactangle.

Does that make sense? Write again if you need more help.

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