Calculate ellipse from 4 points?

In summary, the conversation discusses a professional statistician's question on finding the maximum possible eccentricity of a non-perfectly round cylindrical object using only two measurements taken 90° from each other. The individual is seeking assistance in calculating the lengths of the major and minor axes of an ellipse using this limited information. They also mention the possibility of using a coordinate measuring machine or taking multiple measurements to perform a least squares fit. However, it is determined that the problem may be unsolvable due to only having the two distance measurements and not knowing the exact locations of the four points.
  • #1
bww2000
2
0
I have been scouring the Internet and various geometry books trying to figure out an issue I'm dealing with at work (I'm a professional statistician). It's been over 30 years since I've had a geometry class, so my brain is a bit rusty in this area. Here's what I have so far, can someone tell me if I'm on the right track?

Background

We are taking 2 measurements of the diameter of a cylindrical object using a simple caliper. The two measurements are taken 90° from each other. I use the word "cylindrical" loosely, as the objects are not perfectly round. There is some eccentricity.

Problem

What I would like to know is, using only these two measurements, can I calculate the maximum possible eccentricty of the object? I have posed the question as trying to find the major and minor axes of an ellipse. One possibility of course is that I have made my measurements exactly on the major and minor axes. That's easy. But what if my measurements are on two other random lines through the ellipse? Can I use that information to calculate the lengths of the major and minor axes?

I've attached a jpg image to illustrate the scenario, exagerating the lengths. Refer to the picture for what follows.

I know the length of AB and CD. I know they are perpendicular to each other. Obviously, I don't know theta. I think, given the orientation of my two line segments, an infinite number of ellipses could be drawn through those 4 points.

What I'd like to do is assume my major and minor axes are along the X and Y axes and then calculate the lengths of the major and minor axes for various values of theta. For a given theta, I can calculate the coordinates of A, C, D, and B.

Questions

1) Am I making this too complicated for myself? Is there something really simple I'm overlooking?
2) Do I iterate theta only through 45° or all the way to 89°? It seems to me I only need to iterate theta from 1° to 45°.
3) So far, as I attempt to calculate the lengths of the major and minor axes as in 2), I get some places where my results "blow up". I.e., Excel returns an error of some kind. It could be my equations are incorrect. I'm checking them for the umpteenth time.

Any help or direction would be greatly appreciated!

Thank you!
 
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  • #2
Anyone?
 
  • #3
There is no attachment, btw.

Seems to me 5 pieces of information are needed. I don't think the fact that the two measurements are at right angles helps anything. Can you measure the circumference? That plus the 4 pts might be enough. Also, if you had access to a coordinate measuring machine (CMM), you could take many measurements and perform a least squares fit.
 
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  • #4
If I understand you correctly then you can't solve the problem.

4 points could define ellipses with very extreme aspect ratios.

Imagine the 4 points are all at the same radius from the center. And you assume the axes of the ellipse are parallel to the sides of the square whose corners are the points. Then the points could be on a circle, or an ellipse with any aspect ratio.

However, I think you're saying you want to assume that case isn't happening and only allow the ellipse's axes to be aligned in other orientations which aren't 45deg to the two lines connecting the two pairs of opposite points.

If you have 4 points and the orientation angle then that can give you a set of 5 equations by using the equation for an ellipse. That has 5 unknowns so you should be able to solve it uniquely.

But I guess it would surely be numerically unstable. Can you show your equations?

Oh. I just realized you don't actually know the locations of the 4 points, only the two distances between them. That surely makes it always unsolvable. Still, can you post the equations?
 
  • #5

1. How do you calculate an ellipse from 4 points?

To calculate an ellipse from 4 points, you will need to use the equation of a general ellipse: (x-h)^2/a^2 + (y-k)^2/b^2 = 1. The parameters h and k represent the center of the ellipse, while a and b represent the semi-major and semi-minor axes. By plugging in the coordinates of the 4 points into this equation, you can solve for the values of h, k, a, and b, thus determining the equation of the ellipse.

2. Why do you need 4 points to calculate an ellipse?

An ellipse is a 2-dimensional shape that is defined by 5 parameters: the x and y coordinates of the center, and the lengths of the semi-major and semi-minor axes. By using 4 points, you have enough information to determine the values of all 5 parameters and therefore fully define the ellipse.

3. Can an ellipse be uniquely determined by 4 points?

No, an ellipse cannot be uniquely determined by 4 points. This is because there are infinite ellipses that can pass through 4 given points. However, by using additional constraints or information, such as the orientation of the ellipse or the ratio of the semi-major and semi-minor axes, you can narrow down the possible solutions to a unique ellipse.

4. What are the limitations of using 4 points to calculate an ellipse?

One limitation is that if the 4 points are collinear, meaning they all lie on the same line, there will be no unique solution for the ellipse. Additionally, using only 4 points may not accurately represent the shape of the ellipse if the points are not well-distributed around the perimeter of the ellipse.

5. Can the equation of an ellipse be calculated from points that do not lie on the perimeter?

Yes, the equation of an ellipse can be calculated from any 4 points, even if they do not lie on the perimeter of the ellipse. However, the resulting ellipse may not accurately represent the actual shape of the data points. It is generally recommended to use points that are evenly distributed around the perimeter of the ellipse for more accurate results.

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