# Applied Geometry: Determining whether a given arc represents a circle or an ellipse

So I am trying to figure out if there is a formula/method for determining whether a given arc is part of an ellipse, or a circle given a chord length (Cl) and height (H).

Ex. Now the first arc comes from a unit circle.

The second, visually, at least, is obviously an arc portion of an ellipse.

The third though, is a bit more tricky... It actually still is part of an ellipse, but is a bit more difficult to visualize.

Now, intuitively at least, it seems to me that in the case of the circle, the angle of any portion of that arc should always be constant (?)-- Or, at least, the radius will always be constant for any point along the arc.

The second case could be identified if the radius of any point on the line varies (above ? below ?) the constant that one would find in the case of a circle.

But the third case, I am not so sure...

Further, I am kind of wondering whether or not there is a 'minimum chord length' needed to satisfy said conditions. Again, at least intuitively, it seems if the arc is 'too small' it may be too hard to tell (?)

I say 'applied' in this case because I don't have the formulae for said arcs-- Also, presume arc length (Al) is at least first unknown.

I am working from the impression of drawn arcs, so it would be possible for me to find the points, but I'd have to grid out and measure all the points.

Any thoughts would be greatly appreciated.

• For an arc of circle, all normal lines meet at a same point (the centre). This is not true for an ellipse. But in practice all normal lines of an elliptic arc could meet within a very small region. Oct 29, 2022 at 21:46
• Actually, @Intelligentipauca, I thought about your reply, and then wondered 'how many points' to determine a normal (?)-- The obvious answer is 'at least three', and I agree this could serve as a 'unit test'. And also, I am sorry, perhaps I should have asked my question on the 'Physics' forum, because out of millennium, how did Astronomers observe planets around this world and determine 'it is not in a circle'. Maybe they were lazy and said 'this is the whole orbit'. But your thinking gives me more than enough of a 'bite to chew on', for my purposes. Thanks. Oct 30, 2022 at 6:54
• You should read then Kepler's "Astronomia Nova", as he was the first to conjecture an elliptic shape (for the orbit of Mars, in particular). But he had at his disposal many years of observations. Oct 30, 2022 at 7:39
• @Intelligentipauca the only 'crappy thing' here is I have neither 'years to study', nor an intractable body in motion-- just hand line drawings off a machine (to give you a 'simple' idea', it is any number of screen printing machines-- I mean I think these must be completely circular in their motion, but somehow 'if not', my 'big idea' is in a lot of trouble, or just at least can work, just with some refinement). I ask the question, just to 'get out of trouble', before it begins, in a sense. Oct 30, 2022 at 14:16
• The measurements you're taking do not determine the answer to your question: for any three noncolinear points, there is a circle that passes through them, and there are also non-circular ellipses that pass through them.
– JBL
Oct 31, 2022 at 15:20

... arc is part of an ellipse, or a circle given a chord length (Cl) and height (H).

however, a comment from JBL indicates that those two measurements alone are not enough to answer your question. There are alternative measurements that are enough. Here is one example.

In any circle, if four points $$\;A,B,C,D\;$$ (in that order) on the circle are equidistant (i.e., the chord lengths $$\;AB, BC, CD\;$$ are all equal) then also chords $$\;AC\;$$ and $$\;BD\;$$ have the same length. If not, then the curve is not a circle. Thus, place four equidistant points on your curve and measure the chord lengths $$\;AC\;$$ and $$\;BD.\;$$ If the lengths are different, then the curve is not a circle. It may or may not be an ellipse or any other kind of curve. For more assurance, repeat this several times with different starting point $$\;A\;$$ and chord lengths, and see if you get equality of chord lengths.

• This is a very good answer; a simple construction + easy-to-compute test quantity should keep the inherent experimental and measurement errors under control.
– JBL
Nov 2, 2022 at 11:47
• There is another fact : it is surely impossible to distinguish an arc of an ellipse with small eccentricity and an arc of circle. Mar 6 at 18:37

You don’t show any values. But. A vertical line from the highest point, running through line h. Then a chord from the point where your chord and the height intersect the figure. A perpendicular line from that chord’s midpoint. You now have the center of a circle, to check if in fact it’s a circle. If it’s equidistant from any other point, circle. If not, ellipse.

Three points determine a circle. Your diagrams show three points, the ends of the arrows, though the second one will not fit a circle. If you have a fourth point and they are exact you can just ask if it lies on the circle defined by the other three or not. If the points are not exact you will not be able to tell a circle from an ellipse with small eccentricity. You can do the same as the first point-ask if the fourth point is within a certain distance from the circle. If it is, you can do a numeric least squares to find the best fit circle. If not, you can declare that it is not a circle and find an ellipse that fits.