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Given a certain curve, or part of a circle, how do you express how much it is bent?

If I have for example 1/3 of a circle (without seeing the full circle). How do I calculate and express that it is bent in a way that it forms 1/3 of a circle.

And as a follow-up on this question, how can I then construct the full circle, from this data.

Excuse me if it is unclear, it's hard for me to explain this in English.

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What information do you have available to you? If it is feasible for you to read off the first and second derivative of the curve you can see, then you can find the radius of the circle (See en.wikipedia.org/wiki/Curvature#Curvature_of_a_graph) which is then enough to draw the full circle. –  Ragib Zaman Sep 26 '11 at 15:02
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To repeat the comment of @Ragib Zaman, what does given mean here? There is the Euclidean geometry construction, pick two points on the given part. Find the perpendicular bisector. Do this again with two other points. The two perpendicular bisectors meet at the centre of the circle. –  André Nicolas Sep 26 '11 at 15:16

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If you subtract the angles at endpoints of the arc, you will have the length of the arc in degrees. If, like Andrè said, you use the intersection of two perpendicular bisectors to find the center of the arc, you can get the angle and length of the arc as well as any other information on the arc or circle. Then, if you need to express the rate of curvature (as is done in surveying, for example) you can.

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Thank you :) How do you express the rate of curvature? In degrees? I don't have any background in this, I was just wondering how this works, while sitting in the bus and seeing a curve. I thought "how do I know how curved it is, and how large the circle is" –  Simon Verbeke Sep 26 '11 at 15:33
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Sometimes in a survey of a plot of land, they will use a radius and distance along the arc. You can also express the rate of curvature in degrees per distance along the arc. I think the radius is more commonly used to express the rate of curvature in an arc. –  xpda Sep 26 '11 at 15:53

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