# Thread: Define ellipse given 3 of its points

1. ## 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!

2. 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.

3. 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?

4. 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.

5. 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