# Define ellipse given 3 of its points

• Nov 15th 2010, 06:07 AM
ionymous
Define ellipse given 3 of its points
Long story short, I have a camera sensor that detects 3 points a reports them as x,y pairs. All x and y values are positive.

It's been a long time since I've done geometry and algebra and I'm struggling with getting started. Maybe someone can help me.

I know the points are on an ellipse, and I know the major axis is vertical. I don't know the foci, but don't need them in the end.
Ultimately, I need to find the center of the ellipse.

Can this be done with algebra?
Thanks for any help!
• Nov 15th 2010, 11:26 AM
Opalg
Quote:

Originally Posted by ionymous
Long story short, I have a camera sensor that detects 3 points a reports them as x,y pairs. All x and y values are positive.

It's been a long time since I've done geometry and algebra and I'm struggling with getting started. Maybe someone can help me.

I know the points are on an ellipse, and I know the major axis is vertical. I don't know the foci, but don't need them in the end.
Ultimately, I need to find the center of the ellipse.

Can this be done with algebra?
Thanks for any help!

Three points are not enough to determine the ellipse. You need either a fourth point or some additional information such as the eccentricity of the ellipse. There will be a whole family of ellipses, with major axis vertical, through three points, one for each value of the eccentricity from 0 to 1. One of the limiting cases is eccentricity 0, when you get a circle (it is always possible to find a circle through three points, unless they happen to be collinear). The other limiting case, eccentricity 1, is when the ellipse becomes a parabola.

The position of the centre of the ellipse will depend on the eccentricity, so you can't pin that down either, without some extra information.
• Nov 15th 2010, 12:41 PM
ionymous
Thank you very much for your response!
I am struggling a bit with some of the terminology. I've googled about eccentricity and now have a very basic initial understanding.

But I'm basically struggling with the following...
I have convinced myself that if I draw 3 points on a paper, I can draw different ellipses that intersect these points. But I can only ever come up with a single ellipse that has a major vertical axis. That makes me think there must be a mathematical way to determine this ellipse from those 3 points.

What am I missing?
• Nov 15th 2010, 01:15 PM
Opalg
Here's a simple illustration of three ellipses and a circle, all going through the same three points. I have made the major axes horizontal for convenience, but you can obviously rotate the picture to make them all vertical.
• Nov 15th 2010, 06:01 PM
ionymous
I see it now! Thanks!
But I still think there might be an equation I can use.
Looking at the picture made me think... I don't actually want the center point of the ellipse, I just want the X position of the major vertical axis.
In the picture, every ellipse has the same X position for the major vertical axis, but is that just because of where the 3 points were chosen?
What if the three points were in less symetrical locations?... if that makes sense