Ellipse inscribed in a quadrilateral

Hi,

I am trying to built the ellipse inscribed in a convex quadrilateral and failing to find a clear equation for this (or a geometric construction process).

I know that if we have a four-sided convex poligon in the xy plane there will be more than one ellipse tangent to all sides of the quadrilateral and their centers will lie on the line connecting the midpoints of the diagonals.

But what is the equation that describes the unique ellipse (with maximum area) inscribed in any quadrilateral? How can I find the centre location, axis directions and lenghts?

Hope for some mathematical help! ;-)

many thanks

chiara

Attachments:

inscribed ellipse.gif, 6 KB

Replies to This Discussion

Permalink Reply by morteza farhadian dehkordi on December 20, 2010 at 8:46am

check out this

http://chrisjones.id.au/Ellipses/ellipse.html

Permalink Reply by kiara on December 20, 2010 at 10:02am

very helpful indeed!

it's exactly what i was looking for! many thanks,

Permalink Reply by Chris Tietjen on December 20, 2010 at 9:27am

This could be a step towards what you're looking for. Just looking at it it appears that by a scaling factor it might be the max area inellipse of the parallelogram formed by mirroring the triangle across it's longest side. Not a solution for all quads but perhaps a hint as to how to get there.

Steiner Inellipse

Chris

Permalink Reply by Chris Tietjen on December 20, 2010 at 9:47am

That should be 'mirroring and flipping'.

Chris

Permalink Reply by kiara on December 20, 2010 at 10:06am

For sure the Steiner Inellipse is the underlying principle for inscribing an ellipse to a quadrilateral... but in my case is not enough for constructing them as I have all irregular and different quads where no symmetry exists. Thanks anyway!

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