Define Circle Knowing Two Points and ARC LENGTH Only.

AI Thread Summary
The discussion focuses on determining the equation of a circle given two points and the arc length between them, relevant in bending beam scenarios. The user acknowledges the existence of two solutions based on curvature direction. Suggestions include using Newton's approximation method and Taylor expansion for calculations. Additionally, it's noted that knowing both the arc length and the chord length is crucial for solving the problem. The conversation emphasizes the mathematical relationships involved in defining the circle's parameters.
Ryank
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I am having trouble doing exactly what the title says. I have two points and the arc length between them (this is a bending beam type of a situation). Essentially I know where the ends of the beam are and how long the bent beam is and I need to get the equation of the circle. And yes I realize there would be two distinct solutions depending on positive of negative curvature. Any help would be amazing, thanks!
 
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Ryank said:
I am having trouble doing exactly what the title says. I have two points and the arc length between them (this is a bending beam type of a situation). Essentially I know where the ends of the beam are and how long the bent beam is and I need to get the equation of the circle. And yes I realize there would be two distinct solutions depending on positive of negative curvature. Any help would be amazing, thanks!

What is the context of the question? Could you post a sketch? Thanks.
 
There doesn't really need to be a context. You know the length of a chord on the circle and the length of the arc segment between them. I did some digging and did finally manage to find two solutions. One uses Newton's approximation method and the other uses and taylor expansion:

http://norman.rasmussen.co.za/24/radius-from-arc-and-chord-length/
http://www.mathforum.com/dr.math/faq/faq.circle.segment.html#1
 
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Looks like case 1 from Dr. Math that requires Newton's Method is the most straightforward.
 
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Perhaps this sketch will help, it's basic trigonometry.

Since you have fixed two points on the circumference you not only know the arc length you also know the chord length.
 

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